A Mathematical Companion to Quantum Mechanics - Sternberg

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A Mathematical Companion to Quantum Mechanics

Copyright Copyright © 2019 by Shlomo Sternberg All rights reserved. Bibliographical Note A Mathematical Companion to Quantum Mechanics is a new work, first published by Dover Publications, Inc., in 2019. Library of Congress Cataloging-in-Publication Data Names: Sternberg, Shlomo, author. Title: A mathematical companion to quantum mechanics / Shlomo Sternberg (professor of mathematics emeritus, Harvard University). Description: Mineola, New York : Dover Publications, Inc., 2019. Identifiers: LCCN 2018044258 | ISBN 9780486826899 | ISBN 0486826899 Subjects: LCSH: Quantum theory. | Mathematical physics. Classification: LCC QC174.12 .S8445 2019 | DDC 530.12—dc23 LC record available at https://lccn.loc.gov/2018044258 Manufactured in the United States by LSC Communications 82689901 2019 www.doverpublications.com

To Aviva Green

Table of Contents 1

Introduction

2

The Fourier Transform 2.0.1 Conventions 2.1 Basic Facts about the Fourier Transform Acting on S 2.1.1 The Fourier Transform on L2 2.2 Tempered Distributions 2.3 The Laplace Transform 2.3.1 The Mellin Inversion Formula

3

The Spectral Theorem, I 3.1 The Exponential Series for a Bounded Operator 3.2 The Functional Calculus A → f (A) 3.3 Gelfand’s Formula for the Spectral Radius 3.3.1 Normal Operators 3.3.2 The Spectrum of an Element in an Algebra 3.3.3 The Spectral Radius 3.3.4 Gelfand’s Proof of the Spectral Theorem for Bounded Selfadjoint Operators 3.4 Non-negative Operators

4

Unbounded Operators 4.1 Unbounded Operators 4.1.1 Linear Operators and Their Graphs 4.1.2 The Resolvent and the Resolvent Set 4.1.3 The Adjoint of a Densely Defined Linear Operator

5

Semi-groups, I 5.1 The Bounded Case 5.1.1 The Resolvent from the Semi-group 5.1.2 The Semi-group from the Resolvent 5.1.3 The Two Resolvent Identities 5.2 Sectorial Operators 5.2.1 Definition of et A When A Is Sectorial 5.2.2 The Integral Converges and Is Independent of r 5.2.3 The Semi-group Property 5.2.4 Bounds on et A 5.2.5 The Derivatives of et A, Its Holomorphic Character 5.2.6 The Limit of et A as t → 0+ 5.2.7 The Resolvent as the Laplace Transform of the Semi-group

6

Self-adjoint Operators 6.1 Recall about Unbounded Operators and Their Spectra 6.2 The Spectrum of a Self-adjoint Operator Is Real

7

Semi-groups, II 7.1 Equibounded Continuous Semi-groups 7.1.1 The Infinitesimal Generator

8

Semi-groups, III 8.1 Using the Mellin Inversion Formula 8.2 The Hille-Yosida Theorem 8.2.1 Contraction Semi-groups 8.3 The Spectral Theorem, Functional Calculus Form 8.3.1 The Functional Calculus for Functions in S 8.4 The Dynkin-Helffer-Sjöstrand Formula 8.4.1 Symbols 8.4.2 Slowly Decreasing Functions 8.4.3 Stokes’ Formula in the Plane

8.4.4 Almost Holomorphic Extensions 8.4.5 The Dynkin-Helffer-Sjöstrand Formula 8.4.6 The Dynkin-Helffer-Sjöstrand Formula for the Resolvent 8.4.7 Davies’ Proof of the Spectral Theorem 8.5 Monotonicity 8.6 Dissipative Operators, the Lumer-Phillips Theorem 8.7 The Key Points of the Preceding Chapters 9

Weyl’s Theorem on the Essential Spectrum 9.1 An Example, the Square Well in One Dimension 9.2 The Kato-Rellich Theorem, the Notion of a Core 9.3 The Spectrum as “Approximate Eigenvalues” 9.3.1 An Alternative Approach Using the Born Approximation 9.3.2 There Is No Spectrum of the Square Well Hamiltonian below −K 9.4 The Discrete Spectrum and the Essential Spectrum 9.4.1 Characterizing the Essential Spectrum 9.4.2 A Slightly Different Characterization of the Essential Spectrum 9.5 The Stability of the Essential Spectrum 9.5.1 Preview: How We Will Use Weyl’s Theorem 9.5.2 Proof That A + B Is Self-adjoint 9.5.3 Proof That the Essential Spectrum of A Is Contained in the Essential Spectrum of A + B 9.5.4 Proof That the Essential Spectrum of A Contains the Essential Spectrum of A + B 9.6 The Domain of H0 in One Dimension 9.7 Back to the Square Well (in One Dimension) 9.7.1 The Eigenvalues for the Square Well Hamiltonian 9.7.2 Tunneling

10 More from Weyl’s Theorem 10.1 Finite Rank Operators

10.1.1 A Compact Operator Is the Norm Limit of Finite Rank Operators 10.2 Compact Operators 10.3 Hilbert Schmidt Operators 10.3.1 Hilbert Schmidt Integral Operators 10.3.2 A More General Definition of Hilbert Schmidt Operator 10.3.3 An Important Example 10.4 Using Weyl’s Theorem for H0 + V 10.4.1 Applications to Schrödinger Operators 10.5 The Harmonic Oscillator 10.5.1 Weyl’s Law 10.5.2 Verifying Weyl’s Law for the Harmonic Oscillator 10.6 Potentials in L2 ⊕ L∞ in R3 10.6.1 The Hydrogen Atom 11 Extending the Functional Analysis via Riesz 11.1 Integration According to Daniel 11.2 A Riesz Representation Theorem for Measures 11.3 Verifying That f ↦ O(f) Is a Functional Calculus 12 Wintner’s Proof of the Spectral Theorem 12.1 Self-adjoint Operators and Herglotz Functions 12.1.1 Herglotz Functions 12.1.2 The Borel Transform or the Stieltjes Transform 12.2 The Measure μx.x 12.2.1 The Measure μx.y 12.3 The Spectral Theorem via Wintner 12.4 Partitions of Unity or Resolutions of the Identity 12.5 Stone’s Formula and the Stieltjes Inversion Formula 13 The L2 Version of a Spectral Theorem 13.1 The Cyclic Case

13.2

13.3

13.4

13.5

13.1.1 Cyclic Vectors 13.1.2 The Isometry U in the Cyclic Case 13.1.3 The General Case Fractional Powers of a Non-negative Operator 13.2.1 Non-negative Operators 13.2.2 The Case 0 < λ < 1 13.2.3 The Case λ = , the Quadratic Form Associated with a Nonnegative Self-adjoint Operator The Lax-Milgram Theorems 13.3.1 Gelfand’s Rigged Hilbert Spaces 13.3.2 The Hermitian Case—Lax-Milgram-III Semi-bounded Operators and the So-called Friedrichs Extension 13.4.1 Semi-bounded Operators 13.4.2 The Friedrichs Extension Hardy’s Inequality and the Hydrogen Atom 13.5.1 Hardy’s Inequality

14 Rayleigh-Ritz 14.1 The Rayleigh-Ritz Method 14.1.1 The Hydrogen Molecule 14.1.2 The Heitler-London “Theory” (1927) 14.1.3 Variations on the Variational Formula 14.1.4 The Secular Equation 14.2 Back to Chemistry: Valence 14.2.1 Two-dimensional Examples 14.2.2 The Hückel Theory of Hydrocarbons 15 Some One-dimensional Quantum Mechanics 15.1 Introduction 15.2 The Sturm Comparison Theorem and Its Consequences 15.2.1 The Sturm Oscillation Theorem 15.2.2 A Dichotomy

15.3 15.4 15.5 15.6

15.2.3 Comparison of Two Equations 15.2.4 Application to Eigenvectors Using the Quadratic Form H (f, g) and Rayleigh-Ritz 15.3.1 The Quadratic Form H(f, g) Perron-Frobenius-Krein-Rutman Variational Principles, Glazman’s Lemma The Zeros of Eigenvectors in Higher Dimensions

16 More One-dimensional Quantum Mechanics 16.1 Introduction 16.2 Some Computations in One Dimension 16.2.1 Motion under the Free Hamiltonian, Spreading of the Wave Packet 16.2.2 The Resolvent of the Free Hamiltonian in One Dimension 16.3 Some Elementary Scattering Theory in One Dimension 16.3.1 On the Far Field Behavior of Solutions of the Wave Equation in One Dimension 16.3.2 The Free Problem 16.3.3 The Jost Solutions 16.3.4 A Brief Excursion about Wronskians 17 Some Three-dimensional Computations 17.1 The Yukawa Potential in Three Dimensions 17.1.1 The Yukawa Potential and the Resolvent 17.1.2 The Time Evolution of the Free Hamiltonian 18 Bound States and Scattering States 18.1 Introduction 18.2 The Mean Ergodic Theorem 18.3 Recall: The Kato-Rellich Theorem 18.4 Characterizing Operators with Purely Continuous Spectrum 18.5 The RAGE Theorem 18.5.1 The Spaces M0 and M∞

18.5.2 Using the Mean Ergodic Theorem 18.5.3 The Amrein-Georgescu Theorem 18.6 Kato Potentials 18.7 Ruelle’s Theorem 19 The Exponential Decay of Eigenstates 19.1 Introduction 19.2 The Agmon Metric 19.2.1 The Agmon Distance 19.3 Agmon’s Theorem 19.3.1 Conjugation and Commutation of H0 by a Multiplication Operator 19.4 Some Inequalities 19.5 Completion of the Proof 20 Lorch’s Proof of the Spectral Theorem 20.1 The Riesz-Dunford Calculus 20.1.1 The Riemann Integral over a Curve 20.1.2 Lorch’s Proof of Gelfand’s Formula for the Spectral Radius 20.1.3 Stone’s Formula 20.2 Lorch’s Proof of the Spectral Theorem 20.2.1 The Point Spectrum 20.2.2 Operators with Pure Point Spectrum 20.2.3 Partition into Pure Types 20.2.4 Completion of Lorch’s Proof 21 Scattering Theory via Lax and Phillips 21.1 Notation 21.1.1 Examples 21.2 Incoming and Outgoing Subspaces 21.3 Breit-Wigner 21.4 Strongly Contractive Semi-groups

21.5 The Sinai Representation Theorem 21.5.1 The Sinai Representation Theorem 21.5.2 The Stone-von Neumann Theorem 21.6 The Stone-von Neumann Theorem 21.6.1 The Heisenberg Algebra and Group 21.6.2 The Stone-von Neumann Theorem 22 Huygens’ Principle 22.1 Introduction 22.2 d’Alembert and Duhamel 22.3 Fourier Analysis 22.3.1 Using the Fourier Transform to Solve the Wave Equation 22.3.2 Conservation of Energy at Each Frequency 22.3.3 Distributional Solutions 22.4 The Radon Transform 22.4.1 Definition of the Radon Transform 22.4.2 Using the Fourier Inversion Formula for the Wave Equation 22.4.3 Huygens’ Principle in Odd Dimensions > 1 23 Some Quantum Mechanical Scattering Theory 23.1 Introduction 23.1.1 Notation 23.1.2 The Spaces M+ and M− 23.1.3 Scattering States and the Scattering Operator 23.2 The Scattering Operator 23.3 Properties of the Wave and Scattering Operators 23.3.1 Invariance Properties of M± and R± 23.3.2 Cook’s Lemma 23.3.3 The Chain Rule 23.4 “Time Independent Scattering Theory” 23.4.1 Expressing the Wave Operators in Terms of the Resolvents 23.4.2 A Perturbation Series for the Wave Operators

23.5 Meromorphic Continuation of the Resolvents 23.6 The Lippmann-Schwinger Equation—A Reference 24 The Groenewold-van Hove Theorem 24.1 Poisson Algebra, Notation 24.1.1 The Polynomials of Degree ≤ 2 Form a Maximal Poisson Subalgebra of the Algebra of All Polynomials 24.1.2 The Action of sp(2) on Homogeneous Polynomials 24.1.3 Maximality of the Polynomials of Degree ≤ 2 24.1.4 Expressing q2 p2 in Two Different Ways as Poisson Brackets 24.2 A Change in Notation to Self-adjoint Operators 24.3 Proof of the Groenewold-van Hove Theorem 25 Chernoff’s Theorem 25.1 Convergence of Semi-groups 25.1.1 Resolvent Convergence 25.2 Chernoff’s Theorem 25.2.1 Lie’s Formula 25.2.2 Statement of Chernoff’s Theorem 25.2.3 Proof of Chernoff’s Theorem 25.3 The Trotter Product Formula 25.3.1 Commutators 25.3.2 Feynman “Path Integrals” from Trotter 25.3.3 The Feynman-Kac Formula 26 Some Background Material 26.1 Borel Functions and Borel Measures 26.2 The Basics of the Geometry of Hilbert Space 26.2.1 Scalar and Semi-scalar Products 26.2.2 The Cauchy-Schwarz Inequality 26.2.3 The Triangle Inequality 26.2.4 Pre-Hilbert Spaces

26.2.5 Normed Spaces 26.2.6 Completion 26.2.7 The Pythagorean Theorem 26.2.8 The Theorem of Apollonius 26.2.9 Orthogonal Complements 26.2.10 The Problem of Orthogonal Projection 26.2.11 Orthogonal Projection 26.3 The Riesz Representation Theorem for Hilbert Spaces 26.3.1 Linear and Antilinear Maps 26.3.2 Linear Functions 26.3.3 The Riesz Representation Theorem 26.3.4 Proof of the Riesz Representation Theorem 26.4 The Riemann-Lebesgue Lemma 26.4.1 The Setup 26.4.2 The Averaging Condition 26.4.3 The Riemann-Lebesgue Lemma 26.5 The Riesz Representation Theorem for Measures 26.5.1 Partitions of Unity (in the Topological Sense) and Urysohn’s Lemma 26.5.2 Urysohn Implies Uniqueness 26.5.3 Tentatively Defining the Appropriate σ-Algebra 26.5.4 Countable Sub-additivity of μ in General 26.5.5 Countable Additivity for Elements of ΩB 26.5.6 Proof That ΩB Is a σ-algebra 26.5.7 A Decomposition Theorem Bibliography Index

A Mathematical Companion to Quantum Mechanics

Chapter 1

Introduction Over the years, I taught “Theory of Functions of a Real Variable” at Harvard many times. In addition to standard material such as functional analysis, measure, and integration theory, I included elementary mathematics for quantum mechanics. I thought it would be useful to extract this material and gather it together. There are many books on this subject. The closest competitor is the excellent book Mathematical Methods in Quantum Mechanics by Teschl [25], but my approach to many topics is sufficiently different to warrant the effort of organizing this book. Of course, the multivolume book by Reed and Simon [17] is the classic text on the subject, which goes far beyond anything I cover, and is absolutely necessary for any serious student. I don’t deal with the philosophical and mathematical foundations of quantum mechanics for which I refer to Mackey’s classic [16]. This book presents mathematical methods. As this book is assembled from many lectures, it is not in linear order. It should be regarded as a potpourri of various topics. Nevertheless, here is a rough outline: A key ingredient is the spectral theorem for self-adjoint operators. For example, the first proof for possibly unbounded self-adjoint operators occurs in Wintner’s book of 1929 [27], and the subtitle of the book is “Introduction to the Analytical Apparatus of Quantum Theory.” I reproduce the frontispiece to his book at the end of Chapter 12. I know at least eight different proofs of this theorem. I derive the spectral theorem from the Fourier inversion formula, which says that (for nice functions) f,

where denotes the Fourier transform of f. If we replace x by A and write U (t) instead of eitA, this suggests that we define

This defines a “functional calculus” which has nice properties, as we check in Chapter 3. The problem with this approach is how to make sense of eitA. For bounded operators A, we can use the power series for the exponential. But quantum mechanics requires unbounded operators for which the power series makes no sense. In Chapter 4, I explain unbounded operators, and define their resolvents and spectra. In Chapter 6, I give the rather subtle definition of a (possibly unbounded) self-adjoint operator. Chapters 5, 7, and 8 are devoted to the study of semi-groups and their generators, culminating with the celebrated Hille-Yosida theorem. We derive Stone’s theorem on the nature of unitary one parameter groups on a Hilbert space as a special case of the Hille-Yosida theorem. From Stone’s theorem we get the spectral theorem in functional calculus form, at least for continuous functions f vanishing at infinity. In this chapter I also present the important Dynkin-Helffer-Sjöstrand formula for the functional calculus. The main points of these first eight chapters are summarized at the end of Chapter 8. The next big ticket item is Weyl’s 1909 theorem [26] on the stability of the “essential spectrum.” This general theorem is proved in Chapter 9 and worked out in detail for the square well Hamiltonian of elementary quantum mechanics. Chapter 10 involves more discussion on the details of Weyl’s theorem. One important consequence is to a Schrödinger operator of the form H = H0 + V, where H0 is the “free Hamiltonian” and V is a potential. Weyl’s theorem implies that if V (x) → ∞ as x → ∞ then there is no essential spectrum so the spectrum of H consists entirely of eigenvalues of finite multiplicity, while if V (x) → 0 as x → ∞ then the essential spectrum of H is the same as that of H0. In Chapters 11 and 12, I return to the spectral theorem. In the version given in the first eight chapters, the functional calculus is limited to continuous functions vanishing at infinity. We need to extend it to bounded Borel functions. (For those who don’t know what these are, see section 26.1 below.) In Chapter 11, I use the Riesz representation theorem [19] to get the needed extension.

In Chapter 12, I present Wintner’s proof, which gives the functional calculus directly for bounded Borel functions. I did not do the Wintner approach from the start because it depends on the Borel transformand the Stieltjes inversion formula, which are not as well known (or as standard) as the Fourier transform and its inversion formula. Chapter 13 is devoted to the L2 version of the spectral theorem, which says that any self-adjoint operator is unitarily equivalent to multiplication by a real function on a measure space. In contrast to the functional calculus versions, this isomorphism is highly non-canonical, but it allows us to discuss fractional powers of a non-negative self-adjoint operator. Chapter 14 is devoted to the celebrated Rayleigh-Ritz approximation method for eigenvalues, with applications to “chemical theories.” Chapters 15 and 16 are devoted to quantum mechanics in one dimension, with some results going back to the work of Sturm circa 1850. Chapter 17 does some three-dimensional computations. Chapter 18 is devoted to Ruelle’s theorem: It is a truism in atomic physics or quantum chemistry courses that the eigenstates of the Schrödinger operator for atomic electrons are the bound states—the ones that remain bound to the nucleus —and that the “scattering states” which fly off in large positive or negative times correspond to the continuous spectrum. Ruelle’s theorem gives a mathematical justification for this truism. At the other extreme, Agmon’s theorem says that under appropriate conditions the eigenvectors (which correspond to “bound states”) die off exponentially at infinity. Chapter 19 gives a watered down version of Agmon’s theorem. In Chapter 20 I return once more to the spectral theorem, this time giving Lorch’s proof which makes use of a beautifully “complex variable style” calculus due Riesz and Dunford. Chapters 21 to 23 give a smattering of quantum mechanical scattering theory, including a chapter on Hughens’ principle of wave mechanics. Chapter 24 gives the Groenewold-van Hove Theorem, which shows that Dirac’s proposal that “quantization” gives a correspondence between the Poisson bracket of classical mechanics and the commutator bracket of quantum mechanics does not work beyond quadratic polynomials. Chapter 25 returns to the subject of semigroups and presents an important theorem of Chernoff and some of its consequences. Chapter 26 gives some background material. The first two chapters of [21] provide enough background in real variable

theory for this book. In fact, I assume much less and provide a lot of the background, including a “primer” on Hilbert spaces in Chapter 26. I suggest [25] as a good source for background material. Both [21] and [25] are available gratis on the web. I use the “uniform boundedness principle” (also known as the “Banach-Steinhaus Theorem”) several times. A good treatment can be found on page 39 of [25] and on page 98 of [21]. I use the Stone-Weierstrass theorem (see [25] pp. 59–61) several times. I use the Cauchy integral formula and Cauchy’s theorem (which can be found near the opening of any text on complex variable theory) throughout the book. I thank Chen He and Rob Lowry for proofreading the text.

Chapter 2

The Fourier Transform I will start with standard facts about the Fourier transform from which I will derive the spectral theorem for bounded self-adjoint operators in Chapter 3. 2.0.1

Conventions

The space S consists of all complex valued functions on ℝ that are infinitely differentiable and vanish at infinity rapidly with all their derivatives in the sense that

The ∥·∥m,n give a family of semi-norms on S making S into a Fréchet space—that is, a vector space whose topology is determined by a countable family of seminorms. We use the measure

on ℝ and so define the Fourier transform of an element of S by

and the convolution of two elements of S by

2.1

Basic Facts about the Fourier Transform Acting on S

We are allowed to differentiate with respect to ξ under the integral sign since f (x) vanishes so rapidly at∞. We get

So the Fourier transform of Integration by parts (with vanishing values at the end points) gives

So the Fourier transform of

Putting these two facts together gives

Theorem 2.1.1. The Fourier transform is well defined on S and

This follows by differentiation under the integral sign and by integration by parts, showing that the Fourier transform maps S to S. Convolution goes to multiplication.

so

Scaling. For any f ∈ S and a > 0 define Sa f by (Sa) f (x) ≔ f (ax). Then setting u = ax so dx = (1/a)du we have

so

The Fourier transform of a Gaussian is a Gaussian. The polar coordinate trick evaluates

The integral

converges for all complex values of η, uniformly in any compact region. Hence it defines an analytic function of η that can be evaluated by taking η to be real and then using analytic continuation. For real η we complete the square and make a change of variables:

Setting η = i ξ gives

We will make much use of this equation in what follows. Scaling the unit Gaussian. If we set a = ∈ in our scaling equation and define ρ∈ ≔ S∈n so

then

Notice that for any g ∈ S we have (by a change of variables)

so setting a = ∈ we conclude that

for all ∈. Let

and

Then

so

An approximation. Since g ∈ S it is uniformly continuous on ℝ, for any δ > 0 we can find ∈0 so that the above integral is less than δ in absolute value for all 0 < ∈ < ∈0. In short,

The multiplication formula. This says that

for any f, g ∈ S. Indeed the left-hand side equals

We can write this integral as a double integral and then interchange the order of integration, which gives the right-hand side. The inversion formula. This says that for any f ∈ S

Proof. First observe that for any h ∈ S the Fourier transform of as follows directly from the definition. Taking g(x) = multiplication formula gives

is just in the

We know that the right-hand side approaches for each fixed t, and in fact uniformly on any bounded t interval. Furthermore, for all t. So choosing the interval of integration large enough, we can take the

left-hand side as close as we like to small.

by then choosing ∈ sufficiently

Plancherel’s theorem. This says that

Proof. Let

Then the Fourier transform of is given by

so

Thus

The inversion formula applied to

and evaluated at 0 gives

The left-hand side of this equation is

Thus we have proved Plancherel’s formula

The Poisson summation formula. This says that for any g ∈ S we have

Proof. Let

so h is a smooth function, periodic of period 2π and

Expand h into a Fourier series

where

Setting x = 0 in the Fourier expansion

gives

The Shannon sampling theorem. Let f ∈ S be such that its Fourier transform is supported in the interval [−π, π]. Then a knowledge of f (n) for all n ∈ ℤ determines f. This theorem is the basis for all digital sampling used in information technology. More explicitly,

Proof. Let g be the periodic function (of period 2π) which extends the Fourier transform of f. So

and is periodic. Expand g into a Fourier series:

where

or, by the Fourier inversion formula,

But

Replacing n by −n in the sum, and interchanging summation and integration, which is legitimate since the f (n) decrease very fast, this becomes

But

Rescaling the Shannon sampling theorem. It is useful to reformulate this via rescaling so that the interval [−π, π] is replaced by an arbitrary interval symmetric about the origin: In the engineering literature, the frequency λ is defined by

Suppose we want to apply (2.1) to g = Sa f. We know that the Fourier transform

of g is (1/a)S1/a and

So if

we want to choose a so that a2πλc ≤ π or

For a in this range (2.1) says that

or setting t = ax,

The Nyquist rate. This holds in L2 under the assumption that f satisfies supp ⊂ [−2πλc, 2πλc]. We say that f has finite bandwidth or is band-limited with bandlimit λc. The critical value ac = 1/2λc is known as the Nyquist sampling interval and (1/a) = 2λc is known as the Nyquist sampling rate. Thus the Shannon sampling theorem says that a band-limited signal can be recovered completely from a set of samples taken at a rate ≥ the Nyquist sampling rate. 2.1.1

The Fourier Transform on L2

Recall that Plancherel’s formula says that

Define L2(ℝ) to be the completion of S with respect to the L2 norm given by the left-hand side of the above equation. Since S is dense in L2(ℝ) we conclude that

the Fourier transform extends to unitary isomorphism of L2(ℝ) onto itself. The Heisenberg Uncertainty Principle. Let f ∈ (ℝ) with | f (x)|2dx = 1. We can think of as a probability density on the line. The mean of this probability density is

If we take the Fourier transform, then Plancherel says that

as well, so it defines a probability density with mean

Suppose for the moment that these means both vanish. The Heisenberg Uncertainty Principle says that

In other words, if Var(f) denotes the variance of the probability density | f|2 with similar notation for then

Proof. Write −iξ f (ξ) as the Fourier transform of f' and use Plancherel to write the second integral as | f (x)|2dx. Then the Cauchy-Schwarz inequality says that the left-hand side is ≥ the square of

The general case. If f has norm one but the mean of the probability density | f |2 is not necessarily zero (and similarly for its Fourier transform), the Heisenberg uncertainty principle says that

The general case is reduced to the special case by replacing f (x) by

2.2

Tempered Distributions

The topology on S. The space S was defined to be the collection of all smooth functions on ℝ such that

The collection of these norms defines a topology on S which is much finer that the L2 topology: We declare that a sequence of functions {fk} approaches g ∈ S if and only if

for every m and n. A linear function on S which is continuous with respect to this topology is called a tempered distribution. The space of tempered distributions is denoted by S. For example, every element f ∈ S defines a linear function on S by

But this last expression makes sense for any element f ∈ L2(ℝ), or for any piecewise continuous function f which grows at infinity no faster than any polynomial. For example, if f ≡ 1, the linear function associated to f assigns to ϕ the value

This is clearly continuous with respect to the topology of S but this function of ϕ does not make sense for a general element ϕ of L2(ℝ). The Dirac delta function. Another example of an element of S' is the Dirac δfunction which assigns to ϕ ∈ S its value at 0. This is an element of S' but makes no sense when evaluated on a general element of L2(ℝ). Defining the Fourier transform of a tempered distribution. If f ∈ S, then the Plancherel formula implies that its Fourier transform satisfies

But we can now use this equation to define the Fourier transform of an arbitrary element of S': If ℓ ∈ S' we define F (ℓ) to be the linear function

Examples of Fourier transforms of elements of S'. The Fourier transform of the constant 1. If ℓ corresponds to the function f ≡ 1, then

So the Fourier transform of the function which is identically one is the Dirac δfunction. The Fourier transform of the δ function. If δ denotes the Dirac δ-function,

then

So the Fourier transform of the Dirac δ function is the function which is identically one. In fact, this last example follows from the preceding one: If m = F (ℓ) then

But

So if m = F(ℓ) then

where

The Fourier transform of the function x. This assigns to every ψ ∈ S the value

For an element of S we have

So we define the derivative of an ℓ ∈ S' by

Thus the Fourier transform of 2.3

The Laplace Transform

Definition of the (one-sided) Laplace transform. The inversion problem. Let f be a (possibly vector valued) bounded piecewise differentiable function, so that the integral

converges for z with F is called the Laplace transform of f. The inversion problem is to reconstruct f from F. The Laplace transform as a Fourier transform. Let f be a bounded piecewise differentiable function defined on [0,∞). Let c > 0, z = c +i ξ and h the function given by

Then h is integrable and

If the function F were integrable over the line Г given by Fourier inversion formula would say that

, then the

The condition that the function F be integrable over Г, which is the same as the condition that ĥ be integrable over ℝ, would imply that h is continuous. But h will have a jump at 0 (if f (0) ≠ 0). So we need to be careful about the above formula expressing f in terms of F. The philosophy we have been pushing up until now has been to pass to

tempered distributions. But for applications that I have in mind (to the theory of semi-groups of operators) later on in this book, I need to go back to 19th century mathematics—more precisely to the analogue for the Fourier transform of Dirichlet’s theorem about Fourier series: Dirichlet proved the convergence of the symmetric Fourier sum to under the assumptions that the periodic function f is piecewise differentiable. The analogue of the limit of the symmetric sum for the case of an integral is the “Cauchy principal value”:

Theorem 2.3.1. Let h ∈ L(ℝ) be bounded and such that there is a finite number of real numbers a1, . . . , ar such that h is differentiable on (−∞, a1), (a1, a2), . . . , (ar, ∞) with bounded derivative (and right- and left-handed derivatives at the end points). Then for any x ∈ ℝ we have

In the proof of this theorem we may take x = 0 by a shift of variables. So we want to evaluate the limit of

where the interchange of the order of integration in the first equation is justified by the assumption that h is absolutely integrable. The function H is absolutely integrable so we can find a large q depending on but independent of R so that

We will study

We will use the evaluation of the “Dirichlet integral”

There are many ways of establishing this classical result. For a proof using integration by parts, see Wikipedia under “Dirichlet integral.” An alternative proof can be given via a contour integral. In fact, this evaluation will also be a consequence of what follows: it is clear that the integral converges, so if we let k denote the value and carry the k throughout the proof, we will find that since the above formula is a special case of our Laplace inversion formula for the Heaviside function. The proof of the theorem will proceed by integration by parts: Let

We are interested in studying

as R →∞. Now H is piecewise differentiable with a finite number of points of non-differentiablity where the right- and left-handed derivatives exist as in the theorem. Break the integral on the right up into the sum of the integrals over intervals of differentiability. For example, the last integral will contribute

Integration by parts gives

We are assuming that H and Hare bounded. Since s(Ru) →0 as R →∞and s(Rq) →0 as q →∞, the contribution of these terms tend to zero, assuming that p > 0. The same integration by parts argument applies to any interval of the form (a, b) where a ≠ 0. For the interval (0, c) we have

The first and third terms tend to zero as before, and since , proving the theorem.

we are left with

Comment. The hypotheses on h in the theorem are far too strong. In fact, for a scalar valued function h, all that needs to be assumed is that h is in L and is locally of bounded variation. Even weaker hypotheses work. For example, see Widder, The Laplace Transform. 2.3.1

The Mellin Inversion Formula

In any event, we have proved the Mellin inversion formula

where the left-hand side is interpreted to mean and the “contour integral” on the right is interpreted as a Cauchy principal value.

Chapter 3

The Spectral Theorem, I This chapter is devoted to the spectral theorem for bounded self-adjoint operators. 3.1

The Exponential Series for a Bounded Operator

Suppose that B is a bounded operator on a Banach space ℬ. (See section 26.2.6 for more information on a Banach space.) For example, B might be any linear operator on a finite dimensional space. Then the series

converges for any t. (We will concentrate on t real, and eventually on t ≥ 0 whenwe get to more general cases.) Convergence is guaranteed as a result of the convergence of the usual exponential series in one variable. (There are serious problems with this definition from the point of view of numerical implementation, which we will not discuss here.) The standard proof using the binomial formula shows that

Also, the standard proof for the usual exponential series shows that the operator valued function is differentiable (in the uniform topology) and that

In particular, let A be a bounded self-adjoint operator on a Hilbert space (meaning that A∗ = A) and take B = i A. So let

Then Furthermore,

defines a one parameter group of bounded transformations on .

and since U (0)U(0)∗ = I we conclude that

In words: the operators U (t) are unitary. 3.2

The Functional Calculus A→ f (A)

We present for a bounded self-adjoint operator A and f ∈ S. Recall the Fourier inversion formula for functions f ∈ S, which says that

If we replace x by A and write U (t) instead of eitA, this suggests that we define

We want to check that this assignment has the properties that we would expect from a “functional calculus.” This map is clearly linear in f since the Fourier transform is also linear. We now check that it is multiplicative: Checking that (f g)(A) = f (A)g(A). To check this, we use the fact that the Fourier transform takes multiplication into convolution, i.e., that so

Checking that the map sends For the standard Fourier transform, we know that the Fourier transform of is given by

Substituting this into the right-hand side of (3.1) gives

by making the change of variables s = −t. Checking that Define g by

So g is a real element of S and

so

denote the sup norm of f, and let

So by our previous results,

i.e.,

So for any v ∈ we have

proving that

Enlarging the functional calculus to continuous functions vanishing at infinity. The inequality (3.2) allows us to extend the functional calculus to all continuous functions vanishing at infinity. Indeed if is an element of L1 so that its inverse Fourier transform f is continuous and vanishes at infinity (by Riemann-Lebesgue), we can approximate f in the ǁ· ǁ∞ norm by elements of S and so the formula (3.1) applies to f. We will denote the space of continuous functions vanishing at infinity by C0(ℝ). Preview of coming attractions. We will be devoting much space to generalizing this result in two different ways: We will extend the result from bounded to unbounded self-adjoint operators. Of course, this will require us to define “unbounded self-adjoint operators.” In Chapter 11, we will greatly extend the class of functions to which the functional calculus is defined. For example, suppose that we have extended the calculus so as to include functions of the form 1I, where I is an interval on the real line. Since 1I is real valued, we conclude that 1I (A) = 1I (A)∗, i.e., it is selfadjoint. Since 1= 1I, we conclude that 1I (A)2 = 1I (A). In other words, 1I (A) is a self-adjoint projection. We will examine the meaning of the image of this projection. But our first item of business will be to try to understand more deeply the meaning of

and its relation to the resolvent. This will take some work. The method of using the Fourier inversion formula to derive the spectral theorem for bounded self-adjoint operators is not the one as usually taught. I will pause to give a rapid introduction to Gelfand’s proof, which contains much additional information and is of great historical importance. 3.3

Gelfand’s Formula for the Spectral Radius

Preliminaries. Let B be any bounded linear operator on a Hilbert space. I claim that Proposition 3.3.1.

Proof.

so

so

Reversing the role of B and B∗ gives the reverse inequality so

Inserting this into the inequality

gives

so we have the equality 3.3.1

Normal Operators

A (bounded) operator B on a Hilbert space is called normal (a bad but standard definition) if it commutes with its adjoint:

So a self-adjoint operator A is normal as is any polynomial (with possibly complex coefficients) in A. Proposition 3.3.2. If B is normal, then

Proof. Applying Proposition 3.3.1 to B2, then to BB∗, and then once again to B gives

where we used the normality in the second equation. In short, , and hence

3.3.2

The Spectrum of an Element in an Algebra

All algebras will be over the complex numbers, be associative, and have an identity element. The spectrum of an element x in an algebra is the set of all λ ∈ ℂ such that (x −λe) has no inverse where e is the identity element in the algebra. We denote the spectrum of x by Spec(x). We are interested in the case that our algebra is the algebra of all bounded operators on a Hilbert space and denote a typical element of our algebra by T.

The complement of Spec(T) is called the resolvent set ρ(T) of T. So if z ∈ ρ(T) then zI − T has a bounded inverse, which we denote by R(z, T) and call the resolvent of T at z. The resolvent set of a bounded operator is open. Theorem 3.3.1. If z ∉ Spec(T) and c ≔ ║R(z, T)║, then the spectrum does not intersect the disk

For w in this disk valued function of w.

and so is an analytic operator

For the proof, let C denote the series in the theorem so which converges on our disk and

or

showing that C is a left inverse for wI − T. A similar argument shows that it is a right inverse. So we have proved that the series converges to the resolvent proving that the resolvent set is open and hence that the spectrum is closed. The rest of the theorem is immediate. The spectrum of P(x). Proposition 3.3.3. If P is a polynomial then

Proof. The product of invertible elements is invertible. For any λ ∈ C write P(t) − λ as a product of linear factors:

Thus in A and hence (P(x)−λe)−1 fails to exist if and only if (x −μie)−1 fails to exist for some i, i.e., μi ∈ Spec(x). But these μi are precisely the solutions of

Thus λ ∈ Spec(P(x)) if and only if λ = P(μ) for some μ ∈ Spec(x), which is precisely the assertion of the proposition. 3.3.3

The Spectral Radius

If B is a bounded operator on a Banach space, and z ∈ ℂ satisfies

then the series

converges so

exists so z does not belong to the spectrum of B. So if we define |B|sp by

(called the spectral radius of B) then

The spectral radius of Bn. We know that

Applied to Bn and using

we obtain

for any n, and so

Gelfand’s theorem:

. We know that

We claim that the reverse inequality holds with lim inf replaced by lim sup, so that the limit on the right-hand side actually exists and we get the equality

The proof will make use of the uniform boundedness theorem. Proof. Let |μ| < 1/|B|sp so that λ = 1/μ satisfies |λ| > |B|sp and hence I − is invertible. The power series for (I −μB)−1 (the geometric series) is (I −μB)−1 = I −(μB)where

This converges in the open disk of radius However, we know that (I − μB) −1 exists for |μ| < 1/|B| . In particular, for any continuous linear function ℓ on our sp Banach space, the function is analytic and hence (by a theorem in complex variable theory) its Taylor series

converges on this disk. Here we used the fact that the Taylor series of a function of a complex variable converges on any disk contained in the region where it is

analytic. Thus

for each fixed ℓ if |μ| < 1/|B|sp. Considered as a family of linear functions of ℓ, we see that

is bounded for each fixed μ, and hence by the uniform boundedness principle, there exists a constant K = K μ such that

for each μ in the disk, |μ| < 1/|B|sp. In other words,

so

We will give an alternative proof, due to Lorch, of Gelfand’s formula which does not use the uniform boundedness principle, but rather the Riesz-Dunford functional calculus in Chapter 20. Review: Gelfand’s formula for the spectral radius. The spectral radius of a bounded operator B on a Hilbert space is defined by

Gelfand’s formula says that the limit

exists and the spectral radius of B is given by

The spectral radius of a normal operator T is Proof. Recall that we have proved that for a normal operator, This implies that for any k and hence passing to the limit in Gelfand’s formula with n = 2k proves that

when T is normal. 3.3.4 Gelfand’s Proof of the Spectral Theorem for Bounded Self-adjoint Operators Let A be a bounded self-adjoint operator and let B = P(A) for some polynomial P with complex coefficients. We can apply the preceding results to conclude that

where ║P║∞ denotes the sup norm of P restricted to Spec(A), and, recall, Spec(A) is a bounded closed subset of ℝ. The map is an algebra homomorphism. By the Stone-Weierstrass theorem(in fact, by the Weierstrass approximation theorem), this extends to a norm preserving homomorphism of C0(ℝ) to operators on the Hilbert space, where we use the sup norm over Spec(A) on C0(ℝ) and the operator norm on operators. This is (almost) the spectral theorem for bounded self-adjoint operators. In fact, if C0(ℝ) denotes the space of bounded continuous functions on ℝ where now denotes the sup norm on C0(ℝ), then we have proved Theorem 3.3.2. Let A be a bounded self-adjoint operator on a separable Hilbert space, and let L( ) denote the space of bounded linear operators on . There exists a unique linear map from C0(ℝ) to L( ) such that if f (x) = x on Spec(A) then f (A) = A and

3.4

Non-negative Operators

Definition 3.4.1. We say that the bounded self-adjoint operator C is nonnegative (and we write C ≥ 0) if (Cf, f) ≥ 0 for all f ∈ . If A and B are bounded self-adjoint operators we say A ≥ B if A− B ≥ 0. Theorem 3.4.1. If A ≥ I is a bounded self-adjoint operator then A−1 exists and For any f ∈ we have, by Cauchy-Schwarz,

and so

so A is injective, and A−1 (defined on the range of A) satisfies

We must show that the range of A is all of . Proposition 3.4.1. The range of A is dense. Proof. If not, there is an f ∈ with 0 = (f, Ag) = (Af, g) for all g ∈ so Af = 0 and hence f = 0. Proposition 3.4.2. The range of A is all of . Proof. Let g ∈ and choose fn with Afn → g. Since that {fn} is Cauchy. So fn → f for some f ∈ and hence Af = g.

we see

Corollary 3.4.1. If A ≥ 0 then Spec(A) ⊂ [0,∞). Proof. If λ > 0 and B is a self-adjoint operator with B ≥ λI, then it follows the preceding theorem that B−1 exists. Hence (A + λI)−1 exists, so −λ is in the resolvent set of A for any λ > 0. Theorem 3.4.2. For any self-adjoint transformation A we have

Proof. . Similarly, A ≥ −I. ⇐. Since I − A ≥ 0 it follows from the preceding corollary that Spec(A) ⊂ (−∞, 1] and similarly Spec(A) ⊂ [−1,∞) so Spec(A) ⊂ [−1, 1]. So |A|sp, the spectral radius of A is ≤ 1. But |A|sp = ║A║. An immediate corollary of the theorem is the following: Suppose that μ is a real number. Then is equivalent to (μ− ∈)I ≤ A ≤ (μ+∈)I.

Chapter 4

Unbounded Operators 4.1

Unbounded Operators

Up until now we have dealt with bounded operators. Butwe are interested in partial differential operators, such as the heat operator, which (at least when acting on a fixed Banach space) are unbounded. So we must discuss unbounded operators. We will find that for certain types of operators (sectorial operators—see Chapter 5 for the definition) the above discussion about the semi-group generated by an operator goes through with minor modification in the statements and a good bit of work in the proofs. But for more general unbounded operators (as in the Hille-Yosida theorem, which I plan to discuss in Chapter 8) we will have to do major reworking. The direct sum of two Banach spaces. Let B and C be Banach spaces. We make B ⊕ C into a Banach space via

Here we are using {x, y} to denote the ordered pair of elements x ∈ B and y ∈ C so as to avoid any conflict with our notation for scalar product in a Hilbert space. So {x, y} is just another way of writing x ⊕ y. 4.1.1

Linear Operators and Their Graphs

A subspace

will be called a graph (more precisely a graph of a linear transformation) if

Another way of saying the same thing is

In other words, if

then y is determined by x.

The domain and the map of a graph. Let D(Г) denote the set of all

Then D(Г) is a linear subspace of B, but, and this is very important, D(Г) is not necessarily a closed subspace. We have a linear map

The graph of a linear transformation. Equally well, we could start with the linear transformation: Suppose we are given a (not necessarily closed) subspace D(T) ⊂ B and a linear transformation

We can then consider its graph Г (T) ⊂ B ⊕ C which consists of all

Thus the notion of a graph, and the notion of a linear transformation defined only on a subspace of B, are logically equivalent. When we start with T (as usually will be the case) we will write D(T) for the domain of T and Г(T) for the corresponding graph. There is a certain amount of abuse of language here, in that when we write T, we mean to include D(T) and hence Г(T) as part of the definition. Closed linear transformations. A linear transformation is said to be closed if its graph is a closed subspace of B ⊕ C. Let us disentangle what this says for the operator T. It says that if fn ∈ D(T)

then

This is a much weaker requirement than continuity. Continuity of T would say that fn → f alone would imply that T fn converges to T f. Closedness says that if we know that both

then f = lim fn lies in D(T) and T f = g. It is here that we run up against a famous theorem—the closed graph theorem—which says that if T is defined on all of a Banach space B and has a closed graph, then T must be bounded. So if we are considering operators which are not bounded, we have to deal with operators whose domain is not all of B. 4.1.2

The Resolvent and the Resolvent Set

Let T: B → B be an operator with domain D = D(T). A complex number z is said to belong to the resolvent set of T if the operator

maps D onto all of B and has a two-sided bounded inverse. As before, we denote this bounded inverse by R(z, T) or Rz(T) or simply by Rz if T is understood. So

and is bounded. R(z, T) is called the resolvent of T at the complex number z. The complement of the resolvent set is called the spectrum of T and is denoted by Spec(T). Notice that we are “reversing” the order of the definitions: we first define the resolvent set and then define the spectrum as its complement. The spectrum is a closed subset of ℂ. Theorem 4.1.1. The set Spec(T)is a closed subset of C. In fact, if z ∉ Spec(T) and then the spectrum does not intersect the disk

For w in this disk

and so is an analytic operator

valued function of w. Differentiating this series term by term shows that

The proof is as in the bounded case, with some necessary details added. Proof. The series given in the theorem certainly converges in operator norm to a bounded operator for w in the disk. For a fixed w in the disk, let C denote the operator which is the sum of the series. Then

This shows that C maps B to D(T) and has kernel equal to the kernel of R(z, T) which is {0}. So C is a bounded injective operator mapping B into D. Also

which shows that the image of R(z, T) is contained in the image of C and so the image of C is all of D. Now

If f ∈ D and g = (z I − T) f then f = R(z, T)g and so Cg = f −(w−z)C f and hence

or

showing that C is a left inverse for wI − T. A similar argument shows that it is a right inverse. So we have proved that the series converges to the resolvent proving that the resolvent set is open and hence that the spectrum is closed. The rest of the theorem is immediate. Lemma 4.1.1. If T: B → B is an operator on a Banach space whose spectrum is not the entire plane then T is closed.

Proof. Assume that R = R(z, T) exists for some z. Suppose that fn is a sequence of elements in the domain of T with fn→ f and T fn→g. Set hn ≔ (z I − T) fn so

Then R(z f − g) = lim Rhn = lim fn = f. Since R maps B to the domain of T this shows that f lies in this domain. Multiplying R(z f − g) = f by z I − T gives

showing that T f = g. 4.1.3

The Adjoint of a Densely Defined Linear Operator

Suppose that we have a linear operator T: D(T) →C and let us make the hypothesis that

Any element of B∗ is then completely determined by its restriction to D(T). Now consider

defined by

The definition of the adjoint.

Since m is determined by its restriction to D(T), we see that Г∗ = Г(T∗) is indeed a graph. (It is easy to check that it is a linear subspace of C∗ ⊕ B∗.) In other words, we have defined a linear transformation

whose domain consists of all ℓ ∈ C∗ such that there exists an m ∈ B∗ for which ⟨ℓ, T x ⟩ = ⟨m, x⟩ ∀ x ∈ D(T).

The adjoint of a linear transformation is closed. If ℓn → ℓ and mn → m then the definition of convergence in these spaces implies that for any x ∈ D(T) we have

If we let x range over all of D(T), we conclude that Г∗ is a closed subspace of C∗ ⊕ B∗. In other words, we have proved. Theorem 4.1.2. If T: D(T) →C is a linear transformation whose domain D(T) is dense in B, it has a well-defined adjoint T∗ whose graph is given by (4.1). Furthermore, T ∗ is a closed operator.

Chapter 5

Semi-groups, I In this chapter I will discuss the semi-group generated by an operator A, that is the semi-group

I will start with a review of the simplest case of a bounded operator, and then generalize. I plan to cover the Hille-Yosida theorem on semi-groups, derive Stone’s theorem on unitary one parameter groups on a Hilbert space (a theorem which lies at the foundations of quantum mechanics), and the spectral theorem for (possibly) unbounded self-adjoint operators on a Hilbert space, all this by Chapter 8. The key tool that we will use many times in this chapter is the Cauchy integral formula from elementary complex analysis. 5.1

The Bounded Case

Suppose that A is a bounded operator on a Banach space. For example, this can include any linear operator on a finite dimensional space. Then the series

converges for any t. (We will concentrate on t real, and eventually on t ≥ 0 when we get to more general cases.) Convergence is guaranteed as a result of the

convergence of the usual exponential series in one variable. Recall: Identities of etA when A is bounded. The standard proof using the binomial formula shows that

Also, the standard proof for the usual exponential series shows that the operator valued function is differentiable (in the uniform topology) and that

The resolvent set and the resolvent. A point z ∈ ℂ belongs to the resolvent set of the bounded operator A if the operator z I − A has a bounded (two-sided) inverse. Recall that then this inverse is called the resolvent of A and is denoted by R(z, A). So The series for the resolvent when |z| > ║A║.

For example, if z ≠ 0 we have

the geometric series

converges to (I −z−1A)−1. So all z satisfying belong to the resolvent set of A and for such z we have the convergent series expansion

The spectrum. Of course, in finite dimensions, λ ∈ ℂ is by definition an eigenvalue of A if and only if the operator λI − A is not invertible. So in finite dimensions the resolvent set of A consists of all complex numbers which are not eigenvalues. In general, we defined the spectrum to be the complement of the resolvent set. 5.1.1

The Resolvent from the Semi-group

Suppose that

so that z belongs to the resolvent set of A and also the

function

is integrable over (0,∞). We have

So

In words: for the resolvent is the Laplace transform of the one parameter group. If we differentiate both sides of the above equation n times with respect to z,we find that

5.1.2

The Semi-group from the Resolvent

Let Г be a circle of radius

centered at the origin. The claim is that

Proof. From the power series expansion of the resolvent, the contour integral is

By the Cauchy integral formula, unless n = k in which case the integral is 1. So the above expression becomes

If we integrate the equation

by parts we obtain

More generally, integrating by parts n times gives

5.1.3

The Two Resolvent Identities

The first resolvent identity. If z and w both belong to the resolvent set of A, then we can multiply the equation

on the left by R(z, A) and on the right by R(w, A) to obtain

which is known as the first resolvent identity. Its origin has roots in the 19th century, but Wintner, page 143 [27] attributes it to Hilbert and calls it “die Hilbertsche Funktionalgleichung der Resolvente”—“the Hilbert functional equation for resolvents.” We will make much use of this identity. The second resolvent identity. The first resolvent identity relates the resolvents of a single operator A at two different points in its resolvent set. The second resolvent identity, which also has origins in the 19th century, relates the resolvents of two different operators at a point which belongs to the resolvent set of both. Let ϕ and ψ be invertible operators. Clearly

Let A and B be two operators (at the moment both bounded) and let z belong to the resolvent set of both A and B. Apply the above equation to ϕ = R(z, A) and ψ = R(z, B) so as to get the second resolvent identity

The first resolvent identity and Euler’s equation going to use the first resolvent identity in the form

. We are

together with the Cauchy integral formula and our formula for etA in terms of the resolvent to prove that

Proof. For this purpose we choose two circles Г1 and Г2 about the origin of radius with inside (see below) and write

So using (∗) we have

Now the inner integral of the second term vanishes by Cauchy’s theorem since μ is outside the circle . By the Cauchy integral formula the inner integral of the first term is

so the first term gives e(s+t)A as desired. eA · eB ≠ eA+B in general. If A and B are bounded operators, then it is not true in general that eA · eB = eA+B. Indeed, the two sides agree up to terms which are linear in A and B, but the quadratic terms on the left are while the quadratic terms on the right are These do not agree unless AB = BA. So it is not true in general that

Indeed, the Campbell-Baker-Hausdorff formula gives a rather complicated formula for the operator C such that eB = eC · eA. However, consider the following idea of Kantorovitz: Define

and, in general,

In other words, X n looks like the binomial expansion of (B − A)n with all the B’s moved to the left and all the A’s to the right. Then, claim:

Proof. If A and B commute, this is simply the assertion that et B = et (B−A)et A. But in trying to verify (5.2) all the A’s lie to the right of all the B’s, and we never move an A past a B, so (5.2) is true in general. Most of the material for the rest of this chapter (including the figures) is taken (with permission) from [15] and “Analytic Semigroups and ReactionDiffusion Problems, Internet Seminar 2004–2005,” by Luca Lorenzi, Alessandra Lunardi, Giorgio Metafune, and Diego Pallara and from “Introduzione alla teoria dei semigruppi” by Alessandra Lunardi via the Internet. 5.2

Sectorial Operators

A closed operator A is called sectorial if its resolvent set contains a sector S of the form

More precisely, we will assume that there is a positive constant M such that

For example, we will see later that the spectrum of the Laplacian (acting on L2(ℝn)—see later for the precise definition) lies along the negative axis, hence it is the Laplacian sectorial. 5.2.1

Definition of etA When A Is Sectorial

Recall that in the case of a bounded operator A, the exponential series for etA converges, and if is a circle of radius centered at the origin, we proved that

For sectorial operators,we will define etA by a contour integral of etλR(λ, A) over

a modified contour. I think that the idea of doing this goes back to Dunford. Here is the picture of the proposed contour, where we have “opened up” the circle by two branches going off diagonally to the left:

In the figure, we have slightly generalized the notion of a sectorial operator to allow for some positive spectrum, so we assume that the resolvent set contains the set

where ω ∈ ℝ and θ ∈ (2, π). We also assume that

for some positive constant M. We let γr,η be the curve

The curve in the figure is the curve γr,η shifted by ω.

For each t > 0 we define

We must show that: • • • • 5.2.2

This is well defined, i.e., that the integral converges and defines a bounded operator. This has the properties we expect of the exponential, , and (when defined). The definition is independent of the choice of r and η. This coincides with the old definition when A is bounded. The Integral Converges and Is Independent of r

The integral breaks up into three pieces given by

The middle integral causes no convergence problems. For the first and third terms, we know that is bounded by M/r and cosη < 0 so there is no trouble with these integrals. We now show that the integral is independent of the choice of r and η: For this, choose a different r', η' and consider the region D lying between the two curves γr,η + ω and γr'η'+ω lying between them and the cutoff regions Dn ≔ D ∩{|z −ω| < n} for n ∈ ℕ. See the next figure.

The function is holomorphic on the domain Dn and hence its integral over ∂Dn vanishes. The integral over the pieces of the circle |λ−ω| = n lying on the boundary tends to zero by the estimate we gave above, namely that over these integrals the term cos α is negative. This proves that the definition of etA does not depend on the choice of r and η. The bounded case. If A is a bounded operator, we choose and let η → π. The same argument shows that the integral tends to the integral over the circle, as the two radial integrals will tend to terms that cancel one another. So the new definition coincides with the old one for bounded operators. We now go back to general sectorial operators. We define e0A = I. We now prove: Proposition 5.2.1. For all t > 0, etA: X → Dom(A) and for x ∈ Dom(A)

For this purpose, we first prove a lemma about closed operators A:

Lemma 5.2.1. Let I = (a, b) with −∞ ≤ a < b ≤ +∞ and f : I → Dom(A) be a function that is Riemann integrable, and such that the function is also Riemann integrable. Then

Proof. First suppose that I is bounded. Then the assertion S ∈ Dom(A) is true by linearity where S is a Riemann approximating sum

as is the assertion

By assumption,

The fact that A is closed then implies

that

The non-bounded case follows by the same argument since the convergence of the integrals means that Proof of the proposition. Replacing A by A −ωI, we may assume that ω = 0 to simplify the notation. Take f (λ) = etλR(λ, A)x in the lemma: the resolvent maps the entire Banach space X into Dom(A) and hence the lemma implies that et Ax ∈ Dom(A) for any x ∈ X and that

Now by the definition of the resolvent, (λI − A)R(λ, A) = I so AR(λ, A) = λR(λ, A) − I. By Cauchy, so

We have shown that

For x ∈ Dom(A), we have R(λ, A)Ax = AR(λ, A)x so the first equation above shows that for x ∈ Dom(A)

We continue with the assumption that t > 0. We know that etAx ∈ Dom(A) for any x ∈ X and if x ∈ Dom(A) then The lemma applied to the function f (λ) ≔ λetλR(λ, A)x tells us that AetAx belongs to Dom(A). In other words, etAx belongs to Dom(A2) and furthermore

If x ∈ Dom(A), we can move the A past the R(λ, A) to conclude that

If we assume in addition that x ∈ Dom(A2) so that (by definition) Ax ∈ Dom(A), we can apply the proposition to Ax and conclude that

Continuing in this way we conclude Theorem 5.2.1. If t > 0 then etAx ∈ Dom(Ak) for all positive integers k, and all x ∈ X. If x ∈ Dom(Ak) then

Furthermore

5.2.3

The Semi-group Property

The first resolvent identity. Recall that this says that for λ and μ in the resolvent set of A we have:

We are going to use the first resolvent identity in the form

together with the Cauchy integral formula to prove that

For this purpose we write

where η'∈ (η, π/2). So

(The order of integration is irrelevant due to the exponential convergence.) The second integral, where we first integrate over the interior curve, vanishes by Cauchy’s theorem, as the integrand is holomorphic. See the right-hand figure next. Cauchy’s integral formula applied to the inner integrand of the first integral where we integrate over the outer curve first gives

5.2.4

Bounds on et A

Proposition 5.2.2. There is a constant M0 such that

Once again, for the proof we may assume that ω = 0. Also remember that we may replace r by r' in the contour over which we integrate. We may replace r by r/t in the contour for esA, in particular for s = t. We make the change of variables ξ = λt and obtain: Proof.

It follows that

Estimating AetA. Recall that is part of our assumption, and that AR(λ, A) = λR(λ, A)− I so on the path γr,η we have

Hence

Let r →0. The second integral disappears. In the first integral, make the change of variable We get so

Now etA maps X →Dom(A) and for every x ∈ Dom(A) we have AetAx = etA Ax, hence

So applying the above inequality we obtain:

5.2.5

The Derivatives of etA, Its Holomorphic Character

The derivatives of etA for t > 0. I will continue with the harmless assumption that ω = 0. Differentiating the definition

under the integral sign we see that for t > 0 we have

and this is equal to AetA, as we have already seen. So for t > 0 we have

Iterating the formula

gives

for as a map from ℝ+ to bounded operators on X and its derivatives are given as above. In fact, it extends to a holomorphic function in a wedge about the positive xaxis as we shall now see. The holomorphic character of etA. The exponent in the integral along the rays in

where λ = ρeiη and

is

As long as cos(η+τ) < 0, which is the same as this integral converges, and we can differentiate under the integral sign. So if θ is the angle that enters into the sectorial character of A, we see that the function defined by the above integral for some |η| < θ is holomorphic as a function of z for 5.2.6

The Limit of etA as t → 0+

Let x ∈ Dom(A), choose ℝ ∋ ξ > ω so that ξ is in the resolvent set of A, and choose 0 < r < ξ −ω. Let y ≔ ξ x − Ax so that x = R(ξ, A)y. Then by the first resolvent identity

as the second integral vanishes by the Cauchy integral formula as ξ is outside the region to the left of the curve of integration. The term R(A, ξ)/(ξ −λ) in the first integral is of order |λ|−2 for large λ so we can pass to the limit t →0+ under the integral sign to obtain

Now we can cut off the radial pieces of the integral at |λ−ω| = n and close up the curve via the circular arc |λ−ω| = n going from arg(λ−ω) = η to arg(λ−ω) = −η. On this circular arc the integrand is of order |λ|−2 so this portion of the integral goes to zero as n →∞. The point ξ is in the interior of the closed curve. Taking into account that the curve is oriented clockwise rather that counterclockwise, the Cauchy integral formula tells us that the integral equals R(ξ, A)y = x. So we have proved that

for x ∈ Dom(A). We also know that is bounded for 0 < t < 1 by a constant independent of t, so it follows from the above that Proposition 5.2.3. If

then

Conversely, suppose that limt→0+ etAx = y.We know that etAx ∈ Dom(A) so For any ξ in the resolvent set of A we know that R(ξ, A) is a bounded operator which maps all of X into Dom(A). So

So y = x. Proposition 5.2.4. For every x ∈ X and t ≥ 0,

for any t ≥ 0. Remark. Step 2 will be very important for us. Proof. 1. Choose ξ in the resolvent set of A. For any ∈ ∈ (0, t)

Since R(ξ, A)x ∈ Dom(A), the limit of the last term as ∈ → 0 is R(ξ, A)x. So

Since R(ξ, A) maps X into Dom(A), we have proved 1. If we apply ξ I − A to both sides of the above equation we get

which after rearranging the terms says that which is 2. Under the integrability hypothesis, we may move the A inside the integral, which is 3. Proposition 5.2.5. If x ∈ Dom(A) and

then

Conversely, if the limit of the left-hand side exists, call it z, then x ∈ Dom(A) and Ax = z. Proof. By statement 3 of the preceding proposition, we may write the left-hand side of the equation in our proposition as Axds. Since the function is continuous, we can apply the fundamental theorem of the calculus to conclude that the limit is Ax. Conversely: If the limit exists, then etAx → x so both x and z belong to For any ξ in the resolvent set of A, we have

This shows that x ∈ Dom(A). If we apply ξ I − A to both sides of this equation, we get

so

5.2.7

The Resolvent as the Laplace Transform of the Semi-group

Recall the formula obtain

Replace A by A −λI for λ > 0 to

Letting t →∞gives (since etA is uniformly bounded in t > 0)

Applying R(λ, A) to this equation gives

was proved for real positive λ. But by analytic continuation it is true for all λ in the sector S.

Chapter 6

Self-adjoint Operators In Chapters 6–8, we will discuss one parameter semi-groups Tt whose “infinitesimal generator” A is not necessarily sectorial. We want to understand the meaning of the equation

for such operators (where we impose certain conditions on the semi-group Tt or on the operator A). Sectorial operators are not enough due to quantum mechanics. In quantum mechanics, the fundamental object of study is a unitary one parameter group U (t) and Stone’s theorem asserts that all such U (t) are of the form U (t) = etA where A is skew adjoint, i.e., A = i H where H is a self-adjoint operator. Now the spectrum of a self-adjoint operator lies on the real axis (aswe shall prove in a moment), so the spectrum of a skew adjoint operator lies on the imaginary axis. If A is unbounded (as is usually the case in quantum mechanics), then it will not be sectorial. 6.1

Recall about Unbounded Operators and Their Spectra

Recall that if D(H) is a dense subspace of a Banach space B and H : D(H) → C (another Banach space), then it has a well-defined adjoint H ∗ whose graph Г(H)∗ is given by

Furthermore, H ∗ is always closed. We now (temporarily) restrict to the case that is a Hilbert space. We may identify B∗ = C∗ with via the Riesz representation theorem for Hilbert space, which says that the most general continuous linear function on is given by scalar product with an element of . This theorem was published simultaneously in the Comptes Rendus of 1907, compare [5] and [19] where their papers are a few pages apart. So it should (and sometimes is) called the Fréchet-Riesz representation theorem. I will follow standard usage and refer to it as the Riesz representation theorem. The adjoint of an operator on a Hilbert space. If H : is an operator ∗ with D(H) dense in we may identify the graph of H as consisting of all such that

and then write

Notice that we can describe , the graph of H∗, as being the orthogonal complement in of the subspace

Indeed, the condition that {g, h} be orthogonal to M is

i.e., (Hx, g) = (x, h) for all x in D(H) which is precisely the condition for {g, h} to belong to the graph of H ∗. The domain of the adjoint. The domain D of H ∗ consists of those g such that there is an h with (Hx, g) = (x, h) for all x in the domain of H. We claim that D is dense in Suppose not. Then there would be some with (z, g) = 0 for all g ∈ D. Thus . But is the closure of M. This means that there is a sequence xn ∈ D(H) such that Hxn → z and xn → 0. So if we assume that H is closed, we conclude that z = 0. In short, if H is a closed densely defined operator, so is H ∗.

6.2

The Spectrum of a Self-adjoint Operator Is Real

The definition of a self-adjoint operator on a Hilbert space. We now come to a crucial definition: An operator H defined on a domain D(H) ⊂ is called selfadjoint if

The conditions about the domain D(H) are rather subtle. For the moment we record one immediate consequence of the theorem asserting that the adjoint of a densely defined operator is closed. Proposition 6.2.1. Any self-adjoint operator is closed. Symmetric operators. A densely defined operator S on a Hilbert space is called symmetric if

Another way of saying the same thing is: S is symmetric if D(S) is dense and

Every self-adjoint operator is symmetric but not every symmetric operator is self-adjoint. This subtle difference will only become clear as we go along. A sufficient condition for a symmetric operator to be self-adjoint. Let A be a symmetric operator on a Hilbert space . The following theorem will be very useful. Theorem 6.2.1. If there is a complex number z such that A + z I and map D(A) surjectively onto , then A is self-adjoint. Proof. We must show that if ψ and f are such that

then

both

Choose w ∈ D(A) such that

Then for any ϕ ∈ D(A)

Then choose ϕ ∈ D(A) such that (A +z I)ϕ = ψ −w. So (ψ,ψ −w) = (w,ψ −w) and hence , so

Multiplication operators. Here is an important application of the theorem we just proved: Let (X,F,μ) be a measure space and let ≔ L2(X,μ). Since you may not know measure theory at this time, you may take ≔ L2(ℝn). Let a be a real valued F measurable function (say continuous) on X (on ℝn) with the property that a is bounded on any measurable subset of X of finite measure (on bounded subsets of ℝn). Let

Notice that D is dense in . Let S be the linear operator

defined on the domain D. Notice that S is symmetric. Proposition 6.2.2. The operator S with domain D is self-adjoint. Proof. The operator consisting of multiplication by

is bounded since and clearly maps to D. Its inverse is multiplication by i +a. Similarly, multiplication by −i +a maps D onto . So we may take z = i in the previous theorem. Using the Fourier transform. The Fourier transform is a unitary operator on L2(ℝn) (Plancherel’s theorem), and carries constant coefficient partial

differential operators into multiplication by a polynomial. A consequence of a theorem about multiplication operators that we just proved is: Proposition 6.2.3. If D is a constant coefficient differential operator which is carried by the Fourier transform into a real polynomial, then D is self-adjoint. An example is the Laplacian, which goes over into multiplication by under the Fourier transform. The domain of the Laplacian consists of those f ∈ L2 whose Fourier transform have the property that . An even simpler example is the operator We know that the Fourier transform of So we know that the operator is selfadjoint with domain consisting of all f ∈ L2(ℝ) such that the function belongs to L2. (We will find that the spectrum of H is the entire real line.) In quantum mechanics (and elsewhere) one is interested in the “one parameter group” U(t) = e−iHt. For the H as above, this sends which is the Fourier transform of the function. . Now the operator U (t) sending f (x) into f (x −t) is well defined on all of L2 and is unitary. It acts on f by “shifting it” t units to the right. So we can hope that for any self-adjoint operator we can construct U (t) = e−iHt even though −i H is not sectorial, and that the operators U (t) are unitary. This assertion is one half of Stone’s theorem. I plan to prove this in what follows. But then the operator on L2([0, 1]) (with a similar domain) cannot be selfadjoint for the following reason: If it were, it would generate a one parameter group, call it V = V(t). Suppose that f is a (say continuous) function supported on the subinterval [a, b]. Then f cannot tell if it is to be thought of as an element of L2([0, 1]) or of L2(ℝ). So for small values of |t|, we must have (V (t) f (x)) = f (x −t).What happens when the support of f (x −t) crosses the point 1? It cannot simply disappear for then V (t) would not be unitary. So it must reappear at 0. But should it reappear as itself or with a minus sign or by being multiplied by eiθ for some θ? Any such choice would be OK, but this is not determined by H alone. Some “boundary constraints” must be specified. This should demonstrate that the condition of being self-adjoint is rather subtle. The following theorem will be central for us. Once we will have stated and proved the spectral theorem, the following theorem will be an immediate consequence. But we will proceed in the opposite direction, first proving the following theorem and then using it later to prove the spectral theorem:

Theorem 6.2.2. Let H be a self-adjoint operator on a Hilbert space with domain D = D(H). Let

be a complex number with a non-zero imaginary part. Then

is bijective. Furthermore, the inverse transformation

is bounded and in fact

Proof. We will prove this theorem in stages: Let g ∈ D(H) and set f ≔ (cI − H)g = [λI − H]g+iμg. We begin by showing that

We have

The last two terms cancel: Indeed, since g ∈ D(H) and H is self-adjoint we have

since μ is real. Hence

We have thus proved that

We next show that

Indeed, it follows from (6.2) that

for all g ∈ D(H). Since |μ| > 0, we see that f = 0 ⇒ g = 0 so (cI − H) is injective on D(H), and furthermore that (cI − H)−1 (which is defined on im (cI − H)) satisfies

We must show that the image of (cI − H) is all of For this it is enough to show that there is no h ≠ 0 ∈ which is orthogonal to the image of (cI − H). So suppose that

Then

which says that h ∈ D(H∗) and . But H is self-adjoint so h ∈ D(H) and . (Here is where we use the fact that H is self-adjoint.) Thus

Since

, this is impossible unless h = 0.

We show that the image of (cI − H) is all of H, completing the proof of the theorem: Let We know that we can find

The sequence fn is convergent, hence Cauchy, and from

applied to elements of im D(H) we know that

Hence the sequence {gn} is Cauchy, so gn → g for some H is a closed operator. Hence g ∈ D(H) and (cI − H)g = f.

But we know that

Chapter 7

Semi-groups, II After this excursion into self-adjoint operators, I now want to turn to the study of semi-groups and their infinitesimal operators. If Tt is a family of bounded operators on a Banach space B defined for all t ≥ 0 satisfying

and

then we will call Tt a semi-group. Comment. If Tt satisfies Ts+t = Tt ◦ Ts and T0 = I and is strongly continuous at the origin, then it follows from the uniform boundedness theorem that Tt is uniformly continuous at all t ≥ 0 and that there are constants C and M such that

for all t ≥ 0. For the elementary proofs of these facts see the book on semigroups by Kantorovitz [10]. I will not use these facts but simply take a more restricted definition as we will give below. But if we start with the more general definition and multiply Tt by e−Mt we obtain a semi-group whose operators are uniformly bounded. So without much loss of generality we can restrict ourselves to this case. Fréchet spaces. For certain applications (especially to partial differential

equations) it is useful to work in Fréchet spaces (which are a bit more general than Banach spaces) so the next discussion will be devoted to the setting of Fréchet spaces. Recall that a Fréchet space F is a vector space with a topology defined by a sequence of semi-norms and which is complete. An important example is the Schwartz space S as we have seen. Let F be such a space. 7.1

Equibounded Continuous Semi-groups

We want to consider a one parameter family of operators Tt on F defined for all t ≥ 0 and which satisfy the following conditions: • • • •

T0 = I Tt ◦ Ts = Tt+s limt→t0 Tt x = Tt0 x ∀t0 ≥ 0 and x ∈ F For any defining semi-norm p there is a defining semi-norm q and a constant K such that p(Tt x) ≤ Kq(x) for all t ≥ 0 and all x ∈ F.

We call such a family an equibounded continuous semi-group. We will usually drop the adjective “continuous” and even “equibounded” since we will not be considering any other kind of semi-group. The treatment here will essentially follow that of Yosida [28], especially Chapter IX. 7.1.1

The Infinitesimal Generator

We are going to begin by showing that every such semi-group has an “infinitesimal generator,” i.e., can be written in some sense as Tt = eAt. We define the operator A as

That is, A is the operator so defined on the domain D(A) consisting of those x for which the limit exists. Our first task is to show that D(A) is dense in F. The resolvent. For this we begin with a “putative resolvent”

which is defined (by the boundedness and continuity properties of Tt) for all z with Re z > 0. Our experience with bounded or sectorial operators shows that this should be a good candidate for the resolvent of A. One of our tasks will be to show that R(z) as defined in (7.1) is in fact the resolvent of A. We begin by checking that every element of im R(z) belongs to D(A): We have

If we now let h → 0, the integral inside the bracket tends to zero, and the expression on the right tends to x since T0 = I. We thus see that

and

or, rewriting this in a more familiar form,

This equation says that R(z) is a right inverse for z I − A. It will require a lot more work to show that it is also a left inverse. D(A) is dense in F. We will prove that D(A) is dense in F by showing that, taking s to be real, that

Indeed,

for any s > 0. So we can write

Applying any semi-norm p we obtain

For any ∈ > 0 we can, by the continuity of Tt, find a δ > 0 such that

Now let us write

The first integral is bounded by

As to the second integral, let M be a bound for p(Tt x)+ p(x) which exists by the uniform boundedness of Tt. The triangle inequality says that p(Ttx − x) ≤ p(Tt x)+ p(x) so the second integral is bounded by

This tends to 0 as s → ∞, completing the proof that sR(s)x → x and hence that D(A) is dense in F. The differential equation. Theorem 7.1.1. If x ∈ D(A) then for any t > 0

In colloquial terms, we can formulate the theorem as saying that

in the sense that the appropriate limits exist when applied to x ∈ D(A). Since Tt is continuous in t, we have

for x ∈ D(A). This shows that Tt x ∈ D(A) and

To prove the theorem we must show that we can replace by h →0. Our strategy is to show that with the information that we already have about the existence of right-handed derivatives, we can conclude that

As

is continuous, this is enough to give the desired result.

Using Hahn-Banach. So we wish to prove that

To establish the above equality, it is enough, by the Hahn-Banach theorem, to prove that for any ℓ ∈ F∗ we have

In turn, it is enough to prove this equality for the real and imaginary parts of ℓ. So it all boils down to the following lemma in the theory of functions of a real variable: Lemma 7.1.1. Suppose that f is a continuous real valued function of t with the property that the right-hand derivative

exists for all t and g(t) is continuous. Then f is differentiable with f' = g. Proof of the lemma, 1. We first prove that on an interval [a, b] implies that f (b) ≥ f (a). Suppose not. Then there exists an > 0 such that

Set

Then F(a) = 0 and

At a this implies that there is some c > a near a with F (c) > 0. On the other hand, since F (b) < 0, and F is continuous, there will be some point s < b with F (s) = 0 and F(t) < 0 for s < t ≤ b. This contradicts the fact that Proof of the lemma, 2. Thus if m on an interval [t1, t2] we may apply the above result to f (t)−mt to conclude that

and if we can apply the above result to Mt − f (t) to conclude that f (t2)− f (t1) ≤ M(t2 −t1). So if m = min on the interval [t1, t2] and M is the maximum, we have

Since we are assuming that g is continuous, this is enough to prove that f is indeed differentiable with derivative g. We will now conclude that we have constructed the resolvent of A:We have already verified that

maps F into D(A) and satisfies

for all z with Re z > 0. We shall show that for this range of z

that (zI − A)−1 exists, and that it is given by R(z). Suppose that

and choose ℓ ∈ F∗ with ℓ(x) = 1. Consider

By the results above, we know that ϕ is a differentiable function of t and satisfies the differential equation

So

which is impossible since ϕ(t) is a bounded function of t and the right-hand side of the above equation is not bounded for t ≥ 0 since the real part of z is positive. Thus

We have from (7.2) that

and we know that R(z)(z I − A)x ∈ D(A). From the injectivity of z I − A we conclude that R(z)(z I − A)x = x. From (z I − A)R(z) = I we see that zI − A maps im R(z) ⊂ D(A) onto F so certainly z I − A maps D(A) onto F bijectively. Hence

and

Summary of where we are. The resolvent strong limit for Re z > 0 and, for this range of z:

is defined as a

We claim that Theorem 7.1.2. The operator A is closed. Suppose that xn ∈ D(A), xn → x and yn → y where yn = Axn. We must show that x ∈ D(A) and Ax = y. Proof. Set

Since R(1, A) = (I − A)−1 is a bounded operator, we conclude that

From (7.4) we see that x ∈ D(A) and from the preceding equation that (I − A)x = x − y so Ax = y. Application to Stone’s theorem. We now have enough information to prove

one half of Stone’s theorem, namely that any continuous one parameter group of unitary transformations on a Hilbert space has an infinitesimal generator which is skew adjoint. Suppose that U (t) is a one parameter group of unitary transformations on a Hilbert space . We have (U(t)x, y) = (x,U(t)−1 y) = (x,U(−t)y) and so differentiating at the origin shows that the infinitesimal generator A, which we know to be closed, is skew-symmetric:

The resolvents (zI − A)−1 exist for all z which are not purely imaginary, and (zI − A) maps D(A) onto all of the Hilbert space . Writing A = i H we see that H is symmetric and that ±i I + H is surjective. Hence H is self-adjoint. This proves that every one parameter group of unitary transformations has an infinitesimal generator form i H with H self-adjoint. We now want to turn to the other half of Stone’s theorem: Wewant to start with a self-adjoint operator H, and construct a (unique) one parameter group of unitary operators U (t) whose infinitesimal generator is i H. This fact is an immediate consequence of the spectral theorem. But we want to derive the spectral theorem from Stone’s theorem, so we need to provide a proof of this half of Stone’s theorem which is independent of the spectral theorem. We will state and prove the Hille-Yosida theorem and find that this other half of Stone’s theorem is a special case. See the next chapter.

Chapter 8

Semi-groups, III In this chapter we cover three big ticket items: • •

•

8.1

The Hille-Yosida theorem, which gives a necessary and sufficient condition for an operator A to generate an equibounded semi-group. The second half of Stone’s theorem, which asserts that if H is a selfadjoint operator on a Hilbert space then i H generates a one parameter group of unitary transformations U(t) = eiHt, −∞ < t < ∞. A functional calculus version of the spectral theorem, which asserts that we have a map from a suitable class of functions on the real line to bounded operators, where H is a self-adjoint operator. This map is an algebra homomorphism and sends We will derive this version of the spectral theorem from Stone’s theorem. Using the Mellin Inversion Formula

I will begin, however, with an application of the Mellin inversion formula for the Laplace transform. In the preceding chapter we started with an equibounded one parameter semi-group Tt, defined its infinitesimal generator, A, and found that the resolvent of A was given by the Laplace transform

when Earlier we obtained, in the case that A is sectorial, an “inversion formula” expressing Tt in terms of the resolvent:

The contour in the integral could be taken as a large circle centered at the origin in case A was bounded. For A sectorial we “opened up” the circle with rays extending to the left, and had no trouble with the convergence of the contour integral. But for non-sectorial operators, such as unbounded skew adjoint operators whose spectrum lies on the imaginary axis, the best we can hope for in an inversion formula is to take to be a vertical line to the right of the imaginary axis, i.e., a line of the form We will find that such an inversion formula exists, but that there are some subtleties. First, we will only be able to find an “inversion formula” for Tt x, x ∈ D(A). This will be a “strong integral,” i.e., an integral in B of R(z, A)x rather than a uniform integral (in the space of operators) as in the sectorial case. Also, it won’t be an honest improper integral, but rather a “Cauchy principal value.” In more detail: Suppose that Tt be an equibounded semi-group on a Banach space with generator A.We know that when the resolvent of A is given by the Laplace transform

We also know that for x ∈ D(A) the function is differentiable with bounded derivative Tt Ax. So we may apply our Mellin inversion formula for the Laplace transform to conclude that for t > 0 we have

where the contour integral is taken as a Cauchy principal value. 8.2

The Hille-Yosida Theorem

Introduction to the Hille-Yosida theorem. We continue the study of an equibounded semi-group Tt with infinitesimal generator A on a Fréchet space F where we know that the resolvent R(z, A) for Re z > 0 is given by

This formula shows that R(z, A)x is continuous in z. The first resolvent equation

then shows that R(z, A)x is complex differentiable in z with derivative −R(z, A)2x. It then follows that R(z, A)x has complex derivatives of all orders given by

On the other hand, differentiating the integral formula for the resolvent n times gives

where differentiation under the integral sign is justified by the fact that the Tt are equicontinuous in t. Putting the previous two equations together gives

This implies that for any semi-norm p we have

since

Since the Tt are equibounded by hypothesis, we conclude Proposition 8.2.1. The family of operators {(zR(z, A))n} is equibounded in Re z > 0 and n = 0, 1, 2, . . . . Statement of the Hille-Yosida theorem. Theorem 8.2.1 (Hille-Yosida). Let A be an operator with dense domain D(A), and such that the resolvents

exist and are bounded operators for n = 1, 2, . . . . Then A is the infinitesimal generator of a uniquely determined equibounded semi-group if and only if the operators

are equibounded in m = 0, 1, 2 . . . and n = 1, 2, . . . . Idea of the proof. If A is the infinitesimal generator of an equibounded semigroup then we know that the {(I −n−1A)−m} are equibounded by virtue of the preceding proposition. So we must prove the converse. Our proof of the converse will be in several stages. The idea of the proof is to construct bounded operators Jn so that we can form the semi-group via the exponential series and use these semigroups to construct approximations to the desired semi-group generated by A. So we begin by constructing the Jn. The definition of Jn. Set

so Jn = n(nI − A)−1 and so for x ∈ D(A) we have

or

Similarly (nI − A)Jn = nI so AJn = n(Jn − I). Thus we have

Since the Jn are bounded, we can construct the one parameter semi-group via the exponential series. Set s = nt. We can then form e−nt exp(nt Jn) which we can write as exp(tn(Jn − I)) = exp(t AJn) by virtue of (8.1).We expect from

that

This then suggests that the limit of the exp(t AJn) be the desired semi-group. Proof of (8.2). We first prove it for x ∈ D(A). For such x we have (Jn − I)x = n−1 Jn Ax by (8.1) and this approaches zero since the Jn are equibounded. But since D(A) is dense in F and the Jn are equibounded we conclude that (8.2) holds for all x ∈ F. Defining the approximating semi-groups. Now define

We know from our study of the exponential series that

which implies that

Thus the family of operators

is equibounded for all t ≥ 0 and n = 1, 2, . . ..

The converge as n →∞uniformly on each compact interval of t. The Jn commute with one another by their definition, and hence Jn commutes

with

By the semi-group property we have

so

Applying the semi-norm p and using the equiboundedness we see that

From (8.2) which tells us that the Jn Ax → Ax this implies that the converge (uniformly in every compact interval of t) for x ∈ D(A), and hence since D(A) is dense and the are equicontinuous for all x ∈ F. The limiting family of operators Tt are equicontinuous and form a semi-group because the have this property. We still need to prove that the infinitesimal generator of this semi-group is A. Let us temporarily denote the infinitesimal generator of Tt by B. So we want to prove that A = B. Let x ∈ D(A). We know that

We claim that

uniformly in any compact interval of t. Indeed, for any semi-norm p we have

where we have used (8.3) to get from the second line to the third. The second term on the right tends to zero as n →∞ and we have already proved that the first term converges to zero uniformly on every compact interval of t. This establishes (8.4). Now

where the passage of the limit under the integral sign is justified by the uniform convergence in t on compact sets. It follows from that x is in the domain of the infinitesimal operator B of Tt and that Bx = Ax. So B is an extension of A in the sense that D(B) ⊃ D(A) and Bx = Ax on D(A). But since B is the infinitesimal generator of an equibounded semi-group, we know that (I − B) maps D(B) onto F bijectively, and we are assuming that (I − A) maps D(A) onto F bijectively. Hence D(A) = D(B). This concludes the proof of the Hille-Yosida theorem. The case of a Banach space. In case F is a Banach space, so there is a single norm the hypotheses of the theorem read: D(A) is dense in F, the resolvents R(n, A) exist for all integers n = 1, 2, . . . and there is a constant K independent of n and m such that

8.2.1

Contraction Semi-groups

In particular, if A satisfies

condition (8.5) is satisfied, and such an A then generates a semi-group. Under this stronger hypothesis we can draw a stronger conclusion: In (8.3) we now have p = and K = 1. Since we see that under the hypothesis (8.6) we can conclude that

A semi-group Tt satisfying this condition is called a contraction semi-group.

The other half of Stone’s theorem. We have already given a direct proof that if S is a self-adjoint operator on a Hilbert space then the resolvent exists for all non-real z and satisfies

This implies (8.6) for A = i S and −i S giving us a proof of the existence of U(t) = exp(i St) for any self-adjoint operator S, a proof which is independent of the spectral theorem. We have

and U(0) = I, so

Thus the operators U (t) are unitary for all t A similar argument shows that U(t)U(−t) ≡ I. We conclude that the U(t) form a one parameter group of unitary operators. 8.3

The Spectral Theorem, Functional Calculus Form

8.3.1

The Functional Calculus for Functions in S

Recall that the Fourier inversion formula for functions f whose Fourier transform belongs to L1 (say for f ∈ S, for example) says that

If we replace x by H and write U (t) instead of eitH this suggests that we define

We want to check that this assignment

has the properties that we would

expect from a functional calculus. The method mimics what we did above for bounded H: Checking that (f g)(H) = (fH)g(H). To check this we use the fact that the Fourier transformtakes multiplication into convolution, i.e., that so

Checking that the map sends For the standard Fourier transform we know that the Fourier transform of f is given by

Substituting this into the right-hand side of (8.7) gives

by making the change of variables s = −t. Checking that Define g by

So g is a real element of S and

denote the sup norm of f, and let

so

So by our previous results,

i.e.,

So for any

we have

proving that

Enlarging the functional calculus to continuous functions vanishing at infinity. The inequality

allows us to extend the functional calculus to all continuous functions vanishing at infinity. Indeed if is an element of L1 so that its inverse Fourier transform f is continuous and vanishes at infinity (by Riemann-Lebesgue) we can approximate f in the norm by elements of S and so the formula (8.7) applies to f. We will denote the space of continuous functions vanishing at infinity by C0(ℝ). Checking that (8.7) is non-trivial and unique. I claim that we know from the preceding that for z not real the function rz given by

has the property that

is given by an integral of the type (8.7). Indeed, suppose, for example, that z = a −ib, b > 0 so that w = iz has positive real part. We know that for A = i H,

Now

so

where 1[0,∞) is the indicator function of [0,∞), i.e., 1[0,∞)(x) = 0 for x < 0 and = 1 for x ≥ 0. A similar argument works for z with positive imaginary part. The proof of our formula for the resolvent of the infinitesimal generator of an equibounded semi-group involved some heavy lifting but not the spectral theorem. This shows that (8.7) is not trivial. Once we know that rz(H) = R(z, H) the Stone-Weierstrass theorem gives uniqueness. 8.4

The Dynkin-Helffer-Sjöstrand Formula

There is a formula due to Dynkin, Helffer, and Sjöstrand for f (H) when f is a C∞ function of compact support which is very useful for applications. It depends on the concept of an almost holomorphic extension which I will explain below. In fact, Davies [4] uses this formula to derive the spectral theorem. The discussion in this section follows [4] rather closely. 8.4.1

Symbols

These are functions which vanish more (or grow less) rapidly at infinity the more you differentiate them. More precisely, for any real number β we let Sβ denote the space of smooth functions on ℝ such that for each non-negative integer n there is a constant cn depending on f such that

It is convenient to introduce the function

so we can write the definition of Sβ as being the space of all smooth functions f such that

for some cn depending on f and all n. For example, a polynomial of degree k belongs to Sk since every time you differentiate it you lower the degree (and eventually get zero). More generally, a function of the form P/Q where P and Q are polynomials with Q nowhere vanishing belongs to Sk where k = deg P −deg Q. The name “symbol” comes from the theory of pseudo-differential operators. 8.4.2

Slowly Decreasing Functions

Define

For each n ≥ 0 define the norm ║· ║n on A by

If f ∈ Sβ, β n ≥ 1 then so another application of the lemma proves that is independent of the choice of n ≥ 1. Notice that the proof shows that we can choose σ to be any smooth function which is identically one in a neighborhood of the real axis and which is compactly supported in the imaginary direction. 8.4.6

The Dynkin-Helffer-Sjöstrand Formula for the Resolvent

Let w be a complex number with a non-zero imaginary part, and consider the function rw on ℝ given by

This function clearly belongs to A so we can form rw(H). The purpose of this section is to prove that

We will choose the σ in the definition of so that w ∈ supp σ. To be specific, choose σ = ϕ(λ|y|/(x) where y is the imaginary part of w and λ is large enough so that w ∉ supp σ. Choose n = 1 in the definition of Proof. For each real number m consider the region

Ωm consists of two regions. The boundary of each consists of three straight line segments (one horizontal at the top (or bottom) and two vertical) and a parabolic piece at the bottom of the top component and the top of the bottom component. We can write the integral over ℂ as the limit of the integral over as m →∞. By Stokes,

For large enough m, w belongs to one or the other of the two regions. If in the above integral we could replace by rw then an application of the Cauchy integral theorem to one of the regions and the Cauchy integral formula to the other would yield (8.17). So we must show that as m →∞we make no error in replacing by rw. Here goes: On each of the four vertical line segments we have

The first summand on the right vanishes when λ|y| ≤ ⟨x⟩. We can apply Taylor’s formula with remainder to the function in the second term. So we have the estimate

along each of the four vertical sides. So the integral of the difference along the vertical sides is majorized by

Along the two horizontal lines (the very top and the very bottom) σ vanishes and is of order m−2 so these integrals are O(m−1). On the parabolic curves σ ≡ 1 and the Taylor expansion yields

as before. The integrals over each of these curves γ is majorized by

8.4.7

Davies’ Proof of the Spectral Theorem

We now show that the map where f (H) is given by the Dynkin-HefllerSjöstrand formula (8.15), satisfies our conditions for a functional calculus. Most of the details of the proof follow methods we have already seen. First some

lemmas: Lemma 8.4.4. If f is a smooth function of compact support which is disjoint from the spectrum of H then f (H) = 0. Proof. We can find a finite number of piecewise smooth curves that are disjoint from the spectrum of H and that bound a region U which contains supp(f). Then by Stokes,

since vanishes on ∂U. Lemma 8.4.5. For all f, g ∈ A, (fg)(H) = f (H)g(H). It is enough to prove this when f and g are smooth functions of compact support. Proof. The product on the right is given by

where K ⊃ supp supp are compact subsets of C. Apply the first resolvent identity in the form

to the integrand to write the above integral as the sum of two integrals. Using our variant of the Cauchy integral formula (8.11) the two “double integrals” become “single integrals” and the whole expression becomes

Now (fg)(H) is defined as

where h = f g. But implies Lemma 8.4.5.

is of compact support and is O(y2) so Lemma 8.4.3

Lemma 8.4.6. This follows from R(z, H)∗ = R(z, H). Lemma 8.4.7. The proof is identical to the proof of 8.8 in Chapter 3 so need not be repeated. Let C0(ℝ) denote the space of continuous functions which vanish at ±∞ with the sup norm. The algebra A is dense in C0(ℝ) by Stone-Weierstrass and Lemma 8.4.7 allows us to extend the map to all of C0(ℝ). So we have proved: Theorem 8.4.2. If H is a self-adjoint operator on a Hilbert space , there exists a unique linear map from C0(ℝ) to bounded operators on such that

We have proved everything except the uniqueness. But item 4) determines the map on the functions rw and the algebra generated by these functions is dense by Stone-Weierstrass. Furthermore, the map is given by the DynkinHelffer-Sjöstrand formula (8.15) for f ∈ A. To get the full spectral theorem we will have to extend this functional calculus from C0(ℝ) to a larger class of functions, for example to the class of

bounded continuous functions or even to the class of bounded Borel measurable functions. For example, for each real number t we might want to consider the function x ↦ eitx and so use our functional calculus to construct eitH. We have already constructed this one parameter group of unitary transformations directly. I will postpone the discussion of this extension to Chapter 11. Alternatively, we will give a proof of the spectral theorem due to Wintner in Chapter 12 which directly gives a functional calculus for bounded Borel functions. 8.5

Monotonicity

A useful result is the following elementary monotonicity fact: Proposition 8.5.1. If f ∈ C0(ℝ) is a non-negative real valued function, then f (H) ≥ 0. Since f is real valued, so that we know that f (H) is self-adjoint. The assertion f (H) ≥ 0 means that for all (f(H)v, v) ≥ 0. Proof. Since f ≥ 0, we can write

8.6

with g ∈ C0(ℝ). Then

Dissipative Operators, the Lumer-Phillips Theorem

Dissipation and contraction. Let F be a Banach space. Recall that a semi-group Tt is called a contraction semi-group if

and that (8.6) is a sufficient condition on operator with dense domain to generate a contraction semi-group. The Lumer-Phillips theorem to be stated below gives a necessary and sufficient condition on the infinitesimal generator of a semi-group for the semigroup to be a contraction semi-group. It is generalization of the fact that the resolvent of a self-adjoint operator has ±i in its resolvent set. Semi-scalar products. The first step is to introduce a sort of fake scalar product in the Banach space F. A semi-scalar product on F is a rule which assigns a number ⟪x, z⟫ to every pair of elements x, z ∈ F in such a way that

We can always choose a semi-scalar product as follows: by the Hahn-Banach theorem, for each z ∈ F we can find an ℓz ∈ F∗ such that

Choose one such ℓz for each z ∈ F and set

Clearly all the conditions are satisfied. Of course this definition is highly unnatural, unless there is some reasonable way of choosing the ℓz other than using the axiom of choice. In a Hilbert space, the scalar product is a semi-scalar product. Dissipative operators. An operator A with domain D (A) on F is called dissipative relative to a given semi-scalar product ⟪·, ·⟫ if

For example, if A is a symmetric operator on a Hilbert space such that

then A is dissipative relative to the scalar product. The Lumer-Phillips theorem Theorem 8.6.1 (Lumer-Phillips). Let A be an operator on a Banach space F with D(A) dense in F. Then A generates a contraction semi-group if and only if A is dissipative with respect to some (and hence to any) semi-scalar product and

Proof. Recall the Hille-Yoshida condition:

Suppose first that D(A) is dense and that im(I − A) = F. We wish to show that (8.6) holds, which will guarantee that A generates a contraction semi-group. Let s > 0. Then if x ∈ D(A) and y = sx − Ax then

implying

We are assuming that im(I − A) = F. This together with (8.19)with s = 1 implies that R(1, A) exists and

In turn, this implies that for all z with |z −1| < 1 the resolvent R(z, A) exists and is given by the power series

by our general power series formula for the resolvent. So we have

and In particular, for s real and |s −1| < 1 the resolvent exists, and then (8.19) implies that . Repeating the process we keep enlarging the resolvent set ρ(A) until it includes the whole positive real axis and conclude from (8.19) that which implies (8.6). As we are assuming that D(A) is dense we conclude that A generates a contraction semi-group. Conversely, suppose that Tt is a contraction semi-group with infinitesimal generator A.We know that Dom(A) is dense. Let ⟪·, ·⟫ be any semi-scalar product. Then

Dividing by t and letting conclude that Re ⟪Ax, x⟫ ≤ 0 for all x ∈ D(A), i.e., A is dissipative for ⟪·, ·⟫, completing the proof of the Lumer Phillips theorem. Once again, this gives a direct proof of the existence of the unitary group generated by a skew adjoint operator. A useful way of verifying the condition im(I − A) = F is the following: Let A∗ ≔ F ∗ →F∗ be the adjoint operator which is defined if we assume that D(A) is dense. Proposition 8.6.1. Suppose that A is densely defined and closed, and suppose that both A and A∗ are dissipative. Then im(I − A) = F and hence A generates a contraction semi-group. Proof. The fact that A is closed implies that (I − A)−1 is closed, and since we know that (I − A)−1 is bounded from the fact that A is dissipative, we conclude that im(I − A) is a closed subspace of F. If it were not the whole space there would be an ℓ ∈ F∗ that vanished on this subspace, i.e.,

This implies that ℓ ∈ D(A∗) and A∗ℓ = ℓ,which cannot happen if A∗ is dissipative by (8.19) applied to A∗ and s = 1. 8.7

The Key Points of the Preceding Chapters • • • • • • •

• •

Facts about the Fourier transform; in particular the Fourier inversion formula The infinitesimal generator of a (n equicontinuous equibounded) semigroup The resolvent of the infinitesimal generator is the Laplace transform of the semi-group The Hille-Yosida theorem Stone’s theorem about one parameter groups of unitary transformations on a Hilbert space, a special case of the Hille-Yosida theorem The spectrum of a self-adjoint operator is real The spectral theorem, functional calculus form, for continuous functions vanishing at infinity, via Stone’s theorem and the Fourier inversion formula The Dynkin-Helffer-Sjöstrand formula The Lumer-Phillips theorem

Chapter 9

Weyl’s Theorem on the Essential Spectrum This chapter is devoted to computing the spectrum of the Hamiltonian (=selfadjoint operator)

where is the “free Hamiltonian” and “V” means multiplication by the real valued function V. I will mostly choose units such that so as to simplify the (intermediate) formulas. We know that H0 is self-adjoint and that if V is a real valued function which is bounded on bounded sets, multiplication by V is a self-adjoint operator. But is their sum self-adjoint?We will prove that under suitable conditions on V the sum is indeed self-adjoint. This will be a consequence of the Kato-Rellich theorem that we will state and prove below. 9.1

An Example, the Square Well in One Dimension

To fix the ideas, I will study the case where n = 1 and where the “potential” V is identically zero for |x| > a and equal to −K for |x| ≤ a:

We will conclude that the spectrum of the square well operator consists of a finite number (at least one) of negative eigenvalues and a “continuous spectrum” consisting of [0,∞). The eigenvectors for the negative eigenvalues are “bound states” in the sense that they die exponentially for |x| > a. This result is taught near the beginning of any elementary quantum mechanics course, usually by “handwaving” definitions and methods that are unconvincing to mathematicians.

There are a number of mathematical subtleties involved, and one of the purposes of this chapter is to cross some of the t’s and dot some of the i’s, getting a lot of results in the general case. We begin with the important: 9.2

The Kato-Rellich Theorem, the Notion of a Core

Definition 9.2.1. If A is a self-adjoint operator with domain D(A), a subspace D ⊂ D(A) is called a core for A if the closure of the restriction of A to D is A. Definition 9.2.2. Let A and B be densely defined operators on a Hilbert space . We say that B is A-bounded if • •

D(B) ⊃ D(A) and There exist non-negative real numbers a and b such that

Notice that if

then (9.1) holds. On the other hand, if (9.1) holds, then

Writing ab = (a∈)(b∈−1) for any ∈ > 0, we get

Thus (9.1) implies (9.2) with a replaced by a + ∈ and b replaced by b + ∈−1. Thus the infimum of a over all (a, b) such that (9.1) holds is the same as the infimum of a over all (a, b) such that (9.2) holds. This common infimum is called the relative bound of B with respect to A. If this relative bound is 0 we say that B is infinitesimally small with respect to A. In verifying (9.1) or (9.2) it is sufficient to do so for all ϕ belonging to a core of A. The following theorem was proved by Rellich in 1939 and was extensively used by Kato in the 1960s and is known as the Kato-Rellich theorem. Theorem 9.2.1. Let A be a self-adjoint operator and Ba symmetric operator which is relatively A-bounded with relative bound a < 1. Then A+ B is selfadjoint with domain D(A). Notice that if B is a bounded operator, then it is A bounded with a < 1 for any A. So the Kato-Rellich theorem implies that our square well operator is selfadjoint. So at least we would know that its spectrum is real. We know that to prove that the symmetric operator A + B is self-adjoint, it is enough to show that for some μ > 0 we have that Ran(A + B ±iμI) = H. We also proved the following result for any self-adjoint operator A: Let c = λ+iμ be any complex number with non-zero imaginary part (i.e., μ ≠ 0). Then

is bijective. Furthermore the inverse transformation

is bounded with bound

For any μ > 0 and any ϕ ∈ D(A) we have

We may write ϕ = (A + iμI)−1ψ and rewrite the above equality (with ± = +) as

In particular,

We now turn to the proof of the Kato-Rellich theorem. Proof. We have

Substituting ϕ = (A +iμI)−1ψ into

gives

Now a < 1 by assumption. So for μ sufficiently large the operator C ≔ B(A + iμI)−1 has norm < 1 and hence I + C is invertible. So Ran(I+C) = . Also Ran(A + iμI) = since A is self-adjoint. Hence Ran(I + C)(A + iμI) = . But

The same argument with −μ implies that A + B is self-adjoint.

9.3

The Spectrum as “Approximate Eigenvalues”

For a general operator A on a Banach space, we defined its resolvent set to consist of those complex numbers z such that zI − A maps the domain of A onto the entire Banach space, and has a bounded two-sided inverse. We defined the spectrum of A as the complement of its resolvent set, and proved that the spectrum of a self-adjoint operator is real. Which real numbers belong to the spectrum of H0? We examine this question at the Fourier transform side where H0 consists of multiplication by . If z is a real number and z < 0, then the operator is bounded by |z|−1 and maps the entire Hilbert space into the domain of H0. So no negative real number belongs to the spectrum of H0. I will now show that all non-negative real numbers do belong to the spectrum of H0 so that the spectrum of H0 consists of [0,∞). I will use the following useful characterization of the spectrum of a selfadjoint operator as consisting of its “approximate eigenvalues.” Theorem 9.3.1. Let A be a self-adjoint operator on a Hilbert space . A real number λ belongs to the spectrum of A if and only if there exists a sequence of vectors un ∈ D(A) such that

We apply this theorem to the spectrum of multiplication by , we can find, for any> 0, a bounded function u supported in the region with Then So taking = 1/n we get corresponding un as in the theorem. So the theorem implies that [0,∞) is the spectrum of H0. Example: [0,∞) lies in the spectrum of the square well Hamiltonian. Here is the idea: On the interval [a,∞) the equation (H −λI)ψ = 0 becomes

The solutions to this equation are of the form aeiγx +be−iγx. No non-zero function of this form can belong to L2 for λ > 0. (And it is easy to see from this that no such λ can be an eigenvalue of H.) But we can construct “approximate eigenvectors” in the sense of (our as yet unproved) theorem: Let ϕ be a non-zero

smooth function of compact support in [0,∞) with

Then

. Set

and

So

showing that λ is in the spectrum. We now turn to the proof of the theorem. First the easy direction: Proof. By definition, if λ does not belong to the spectrum of A then the operator (λI − A)−1 exists and is bounded, so there is a μ > 0 such that

So if and cannot approach 0. So if a sequence as in the theorem exists, then λ belongs to the spectrum of A. This is the easy direction; it is practically the definition of the spectrum. In the other direction, suppose that no such sequence exists. Lemma 9.3.1. There exists a constant c > 0 such that

Proof. If not, there would exist a sequence vn of non-zero elements of D(A) with

Replacing vn by gives a sequence of unit vector in D(A) with This establishes (9.4). From the inequality (9.4) we conclude that the map λI − A is injective. We can also conclude that its image is closed. For if wn = (λI − A)un, un ∈ D(A) with wn →w, then (9.4) implies that the sequence u n is Cauchy and so converges to some u ∈ . We want to show that u ∈ D(A). For any v ∈ D(A) we have

So u ∈ D((λI − A)∗) = D(A∗) = D(A) since A is self-adjoint. To show that λ is in the resolvent set of A we must show that the image of (λI − A) is all of : Let f ∈ and consider the linear function on the image of λI − A given by

Now and by (9.4), this is So this linear function is bounded on the image of λI − A. But this image, being a closed subspace of as we have just proved, is a Hilbert space in its own right, and so we may apply the Riesz representation theorem to conclude that there is a u in the image of λI − A such that

Since A, and hence λI − A is self-adjoint, this implies that u ∈ D(A) and (λI − A)u = f. 9.3.1

An Alternative Approach Using the Born Approximation

Let A be a self-adjoint operator on a Hilbert space , let ρ denote its resolvent set and σ its spectrum. Finally, let R(z) = R A(z) denote its resolvent that is a holomorphic bounded operator valued function on ρ, which satisfies the first resolvent identity

i.e.,

Substituting this into itself at the second term gives

and continue this way to obtain the Born approximation

The bounded operators Rn(z) converge to a bounded operator if |z −z'| < ║R(z)║ −1 and this limit is the resolvent of A at z. This gives our original proof that z → R(z) is holomorphic on ρ. As this series must converge on any disk around z' contained in ρ we conclude that

We can use (∗) to assert that a point u ∈ σ is an “approximate eigenvalue” of A in the following sense: Corollary 9.3.1. If there is a sequence ψn ∈ D(A) such that then u ∈ σ(A).

and



We prove that u ∈ ρ is impossible: For if so, then

since (∗) tells us that is bounded. Conversely, if u ∈ ∂ρ, (∗) says that there is a sequence of zn → u and corresponding vectors ϕn ∈ with Let ψn ≔ R(zn)ϕn and rescale ϕn so that and

9.3.2

There Is No Spectrum of the Square Well Hamiltonian below −K

Indeed, let us state the result more generally: Proposition 9.3.1. Suppose (in any dimension) the potential V has the property that the operator H = H0 + V is self-adjoint and that there is some constant C such that V ≥ C. Then any λ < C is in the resolvent set of H.

is non-negative and applying Cauchy-Schwarz we get

with What we know so far about the square well: •

All of [0,∞) lies in the spectrum, and there are no eigenvalues there.

•

There is no spectrum below −K.

I will now show that the spectrum lying in [−K, 0) consists of eigenvalues of finite multiplicity, which we will then compute. The idea goes back to a theorem of Hermann Weyl [26] of 1909(!), which asserts that the essential spectrum of a self-adjoint operator (definition in a moment) is unchanged by perturbation by a relatively compact operator. 9.4

The Discrete Spectrum and the Essential Spectrum

We let σ = σ(A) denote the spectrum of the self-adjoint operator A.A point λ ∈ σ is isolated if it is an eigenvalue of finite multiplicity with the property that there is an such that the interval (λ−∈, λ+∈) contains no other points of the spectrum of A. The discrete spectrum of A consists of the set of isolated points of the spectrum and is denoted by σd(A), or simplybyσd when A is fixed in the discussion. The complement of the discrete spectrum in σ(A) is called the essential spectrum of A and is denoted by σess(A) or simply by σess when A is fixed in the discussion. The reason for this name will be clear when we get to Weyl’s

theorem. 9.4.1

Characterizing the Essential Spectrum

Suppose that λ is an isolated point of the spectrum. Let λ denote the eigenspace with eigenvalue λ so λ is finite dimensional by assumption. Decompose into the direct sum

The entire interval (λ−, λ+) lies in the resolvent set of the restriction of A to So the restriction of λI − A to has a bounded inverse. So if ψn ∈ D(A) is a sequence of elements of with

then the components of the ψn must tend to 0. The λ components then form a bounded sequence in a finite dimensional space, and hence we can extract a convergent subsequence. So we have proved one half of the following: Theorem 9.4.1. A point λ belongs to the essential spectrum of a self-adjoint operator A if and only if there exists a sequence ψn ∈ D(A) such that

Indeed, we have proved that if λ does not belong to the essential spectrum then there cannot exist such a sequence, for if λ lies in the resolvent set then (λI − A)−1 is bounded and so no sequence with ║ψn║ = 1 and (λI − A)ψn →0 can exist, while if λ is an isolated point of the spectrum we have just proved that no sequence satisfying the conditions of the theorem can exist. Conversely, suppose that λ lies in the essential spectrum. Wewant to construct a sequence as in the theorem. If λ is an eigenvalue of infinite multiplicity, we can construct an orthonormal sequence of eigenvectors ψn so (λI − A)ψn = 0 and the ψn have no convergent subsequence. Otherwise we can find a sequence of λn lying in the spectrum of A with λn ≠ λ

and λn →λ. So λ−λn ≠ 0 and λ−λn lies in the spectrum of A −λn I and hence by our Theorem 9.3.1 characterizing the spectrum, we can find ψn with

and so (λI − A)ψn → 0. We wish to prove that ψn has no convergent subsequence: Proof. If it did, then passing to the subsequence (and relabeling) we would have ψn →ψ and ψ an eigenvector of A with eigenvalue λ. Then

so

which is impossible since (ψn,ψ) →1. 9.4.2

A Slightly Different Characterization of the Essential Spectrum

It will be convenient to replace the middle condition in Theorem 9.4.1 by a slightly different one: A sequence ψn is said to converge weakly to ψ if for every v ∈

It is easy to see that every bounded sequence in a separable Hilbert space has a weakly convergent subsequence. Indeed, if the ψn are bounded, then for each fixed v, {(ψn, v)} is a bounded sequence of complex numbers by CauchySchwarz.We can then apply Cantor’s diagonalization procedure to pass to a subsequence so that (ψn, v) →(ψ, v) for all v in an orthonormal basis. On the other hand, suppose that ψn ∈ satisfies

Then ψn can have no (strongly) convergent subsequence because the strong limit of any subsequence would have to equal the weak limit and hence = 0 contradicting the hypothesis Our desired alternative characterization of the essential spectrum is Proposition 9.4.1. A λ ∈ ℝ belongs to the essential spectrum of a self-adjoint operator A if and only if there exists a sequence ψn ∈ D(A) such that

Proof. If the first two conditions are satisfied then the ψn cannot have a convergent subsequence, as we have just seen. So the conditions of Theorem 9.4.1 characterizing the essential spectrum are satisfied, and hence λ belongs to the essential spectrum of A. Conversely, if λ belongs to the essential spectrum of A then we can find a sequence ψn satisfying the three conditions of Theorem 9.4.1. The first condition implies that the ψn are uniformly bounded so we can choose a subsequence which converges weakly to some ψ ∈ . Let us pass to this subsequence (and relabel). Since our ψn has no convergent subsequence, there is an > 0 such that

for all n. By the third condition in Theorem 9.4.1

so

Since A is self-adjoint, this implies that ψ ∈ D(A) and Aψ = λψ. Consider the sequence

We have

and

for all

So the first two conditions of

Proposition 9.4.1 are satisfied. So is the third because

since 9.5

The Stability of the Essential Spectrum

We now come to the great theorem [26] of Hermann Weyl. Let A be a selfadjoint operator. An operator B is called compact relative to A or, more simply, A-compact if 1.

D(A) ⊂ D(B) and

2.

If un ∈ D(A) is a sequence with

for some constant C then the sequence Bun has a convergent subsequence. For example, a compact operator is A-compact for any A. Theorem 9.5.1 (Weyl). If A is a self-adjoint and B is a symmetric operator which is A compact then A+ B is self-adjoint and has the same essential spectrum as A. 9.5.1

Preview: How We Will Use Weyl’s Theorem

We will show that if V is a bounded potential of compact support then it is H0 compact and hence H0 + V has the same essential spectrum as H0 which is [0,∞). This allows us to conclude that the negative elements of the spectrum of H0 + V are eigenvalues with finite multiplicity. In this chapter I will prove this fact in one dimension, using special facts about H0 in one dimension. I will discuss the general situation in the next chapter. But we will be able to use the one-dimensional facts to see how to compute the eigenvalues for the square well. But first to the proof of Weyl’s theorem:

9.5.2

Proof That A+ B Is Self-adjoint

We will prove that B is infinitesimally small with respect to A. Then KatoRellich implies that A + B is self-adjoint. Proof. For this we begin by proving that B is A-bounded. More precisely, we claim that there are constants k and k' such that

Indeed, if the first inequality did not hold we would be able to find a sequence of un ∈ D(A) with so Bun has no convergent subsequence. If the second inequality did not hold, we could find a sequence of un ∈ D(A) with By passing to a subsequence we may assume that Bun → u in which case we must have Aun →−u. We also must have un →0, so u = 0. But then Aun →0 and un →0 contradicts the hypothesis So we know that there are constants k and k' such that

We can now prove that B is infinitesimally small with respect to A, i.e., that for any ∈ > 0 there is a K∈ such that

Suppose not. This means that there is some > 0 such that for any n we can find a un ∈ D(A) with and

So un → 0 and hence . By passing to a convergent subsequence, we may assume that Bun →w, and hencew = 0, since (Bun, v) = (un, Bv) → 0 for any v ∈ D(A) because B is symmetric. On the other hand,

So we now know from Kato-Rellich that A + B is self-adjoint with domain D(A).

9.5.3 Proof That the Essential Spectrum of A Is Contained in the Essential Spectrum of A+ B Proof. Suppose that λ ∈ σess(A). Choose ψn as in the proposition. Condition 1 says that the and condition 3, which says that (λI − A)ψn → 0 then implies that is bounded. Since B is A compact, we can pass to a subsequence so that now Bψn converges, say to u. For any ϕ ∈ D(A) we have

by condition 2) which says that ψn converges weakly to 0. Since D(A) is dense in this implies that u = 0. Thus

which is condition 3 of Proposition 9.4.1 for A + B. 9.5.4 Proof That the Essential Spectrum of A Contains the Essential Spectrum of A+ B To prove this, by the preceding result, it is enough to prove that B (and hence −B) is compact relative to A + B. Proof. So suppose that

for some constant C. Then it follows from the second inequality in

that

with C'= (k/k)C, completing the proof of Weyl’s theorem. 9.6

The Domain of H 0 in One Dimension

We will now apply Weyl’s theorem to the square well. For this I will use special

facts about the free Hamiltonian in one dimension: Recall that (in any dimension) we defined the domain of H0 as the set of all u ∈ L2(ℝn) such that its Fourier transform û is such that k2 û(k) ∈ L2 in which case H0u is the inverse Fourier transform of k2 û (k). In one dimension we will see that if u ∈ D(H0) then u “is” a continuous function with a continuous derivative. Since elements of L2 are equivalence classes of functions rather than actual functions, I have to be a bit careful about how I say this. I will borrow language from measure theory and use the phrase “almost everywhere equal to” (which you can safely ignore). Continuity. Lemma 9.6.1. If û and k û (k) are both in L2 then u is (almost everywhere equal to) a continuous function. By the Fourier inversion formula

Apply Cauchy-Schwarz to the right-hand side written in the form

and use the facts that 1/(1+k2) is integrable in one dimension and the function is continuous. Differentiablilty. Theorem 9.6.1. The elements of D(H0) are (almost everywhere equal to) continuously differentiable functions. If û and k2 û(k) are in L2 so is k û(k), so we know from the lemma that u is (almost everywhere equal to) a continuous function. Let

By the argument we gave for the lemma, this is “continuous.” We use the lemma to prove the theorem: Proof. We have

Again by Cauchy-Schwarz

The second factor is finite and independent of x and y and the first factor tends to zero. This shows that the derivative of u equals w which is continuous. Remark: If u ∈ D(H0) so that k2 û ∈ L2, then we can write

and use the fact that (1+k2)−2 is integrable in two and in three dimensions to conclude that any element of D(H0) is (almost everywhere equal to) a continuous function. But the argument for differentiability only works in one dimension. 9.7

Back to the Square Well (in One Dimension)

Since D(H) = D(H0) for the operators of the form H = H0 + V that we shall consider, we will be able to use the continuity of u and u' to prove that V is H0 bounded and to determine the eigenfunctions of H. In all the usual physics texts that I have seen, the continuity of u and u' is taken for granted as “physical property” that is needed for a state. In what follows we will use a familiar argument—that since 0 we have the inequality

A Gårding style estimate. Let us go back to the Fourier inversion formula which tells us that

from which we deduce that for u ∈ D(H0) we have

and from which we deduce that for u ∈ D(H0) we have

Since we conclude that there is a constant C such that for any u ∈ D(H0) we have

Let un be a sequence of elements in D(H0) such that

for some constant k. Then for any bounded interval I on the real line we can choose a subsequence of the un which converge uniformly on I by Arzela-Ascoli. Let V be a bounded (say piecewise continuous or even Lebesgue measurable) function of compact support (meaning that it vanishes outside some bounded interval) I. Let unk be the subsequence as above. Then the uniform convergence of the unk implies that the sequence Vunk converges in L2. We have proved that

The spectrum of the square well lying in [−K, 0). We can now apply Weyl’s theorem to conclude that whatever spectrum exists in [−K, 0) is discrete, i.e.,

consists of isolated eigenvalues of finite multiplicity. I have finally reached the starting point of the discussion as is found in the standard quantum mechanics books! 9.7.1

The Eigenvalues for the Square Well Hamiltonian

Using symmetry. Here is the simplest example of the use of group theory to facilitate the computation of eigenvalues: Let U be a unitary operator on that satisfies U 2 = I. So decomposes as a direct sum

where consists of those which satisfy Uu = u and which satisfy Uu = −u. Indeed, writing

consists of those

gives such a decomposition. If H is a self-adjoint operator which commutes with U then H leaves each of the subspaces invariant. Furthermore, if Hu = λu and we decompose u = u+ + u− according to the above decomposition of then each of the components u± also satisfies Hu± = λu±. A key tool in replacing the two element group ℤ/ℤ2 by a more general compact group is the Peter-Weyl theorem, which you can find in my book “Group Theory in Physics.” We apply our symmetry condition to = L2(ℝ) and U given by

Clearly H0 commutes with U. If V(x) = V(−x) then multiplication by V commutes with U. So, for example, every eigenvector of the Schrödinger operator for the square well potential can be decomposed as above, and so we need only study even eigenvectors (those lying in +) or odd eigenvectors (those lying in −). The eigenvector equation in each region. Let

where−K ≤ λ < 0 and we take the positive square roots. For |x| < a the equation Hu = λu is

while for |x| > a the equation Hu = λu is

For x > a the only solution of (9.8) which belongs to L2 is of the form

The even eigenvectors. The even solutions of (9.7) are of the form

The logarithmic derivative, u'/u of this function at x = a is −α tan αa. The logarithmic derivative of Ce−βx at x = a is −β. These must be equal by the differentiability properties of elements of the domain of H0. So we obtain the condition: The eigenvalues of the even eigenvectors.

To understand the solutions of this equation set

so that

the equation of a circle in the (ξ, η) plane. Equation (9.9) becomes

and we are interested in solutions with positive values of ξ and η. The solutions of this equation are the intersections of the curves (ξ, ξ tan ξ) with the circle (9.10). As ξ varies from 0 to π/2, ξ tan ξ increases from 0 to∞, so the curve η = ξ tan ξ always intersects the circle. This shows that there is always at least one (even) eigenvector. Graphic description of the eigenvalues.

The actual values of E obtained from the intersections must be determined numerically As ξ varies from 0 to π/2 the value of ξ tan ξ increases from 0 to ∞and so (ξ, ξ tan ξ) intersects the circle. From π/2 to π, ξ tan ξ is negative and so makes no appearance in the first quadrant. From π to 3π/2 we get a translate of the original curve. So we get the translates of the original curve by multiples of π in the ξ direction. As soon as nπ exceeds the radius of the circle (9.10), we can no longer have an intersection. This shows that there are only finitely many eigenvalues corresponding to even eigenvectors. In fact, the number of “even” eigenvalues is the integer n such that

Notice also that the continuity of u at a requires that cos αa which shows that each of the eigenvalues has multiplicity one. I will leave the computation of the odd eigenvalues to you as an exercise, or

you can read it in any quantum mechanics book. 9.7.2

Tunneling

Notice that though the eigenvectors are exponentially decaying, they are not zero outside the well. This is in contrast to a classical particle trapped in a well which cannot escape. This phenomenon is known as “tunneling.” The first three eigenfunctions, with ψ1 and ψ3 even and ψ2 odd.

Some comments: We proved by explicit computation that the eigenvectors of the square well potential die exponentially at infinity. The general theory of such exponential decay can be found in the book [1] by my old friend Shmuel Agmon (I met Shmuel in 1956 when I was a post doc at NYU). Asimplified version of the general theory developed by Agmon can be found in Chapter 3 of the excellent book [8]. I will discuss this result in Chapter 19. One can make weaker assumptions on the potential V than those needed to get exponential decay of the eigenvectors but still conclude that they are “bound states” in an appropriate sense. The key result in this direction is due to Ruelle, and an excellent exposition of Ruelle’s result, and a nice generalization of it, can be found in the paper by W.O. Amrein and V. Georgescu in Helvetica Physica Acta 46 (1973) pp. 636–658. I will discuss this in Chapter 18.

Chapter 10

More from Weyl’s Theorem I want to discuss more applications of Weyl’s theorem, this time in general n dimensions. The key results are: •

If V is a potential which tends to zero at infinity then

a result that we proved in the special case of the square well potential in one dimension, and •

If V →∞ at ∞ then σess(H0 + V) = ∅ so the spectrum of σess(H0 + V) consists of eigenvectors of finite multiplicity.

For these results I need some information about compact operators and HilbertSchmidt operators which I will present here. 10.1

Finite Rank Operators

An operator is said to be of finite rank if its range, Ran(K), is finite dimensional, in which case the dimension of its range is called the rank of K. If ψ1, . . . ,ψn is an orthonormal basis of Ran(K) and ψ is any element of then

If we set ϕj ≔ K ∗ψj we can write this last equation as

The elements ϕj = K ∗ψj are linearly independent, for if a jϕj = 0 then (a jψj, K f) = 0 for all which cannot happen unless all the a j = 0 since ψ1, . . . ,ψn is an orthonormal basis of Ran(K). So

is the most general expression of a finite rank operator. Then K∗ is given by

as can immediately be checked, and so the adjoint of a finite rank operator is of finite rank. If K is an operator of finite rank and B is a bounded operator then clearly BK and KB are of finite rank. 10.1.1

A Compact Operator Is the Norm Limit of Finite Rank Operators

Proof. Let K be a compact operator and {ϕj} an orthonormal basis of Let So K = Kn on the space spanned by the first n ϕj’s. Let be the orthocomplement of this space, i.e., the space spanned by ϕn+1, ϕn+2, . . .. So

The numbers are decreasing and hence tend to some limit, ℓ, and so we can find a sequence ψn with and lim The ψn converge weakly to 0, and hence so do the K ψn. We can choose a (strongly) convergent subsequence which must therefore converge to 0, and so ℓ = 0. The converse. We will now derive a converse of the above fact, and in what

follows we will assume that K is the norm limit of finite rank operators. Theorem 10.1.1. Suppose that K is the norm limit of finite rank operators and is self-adjoint. Then σess(K) ⊂ {0} with equality if and only if is infinite dimensional. We must show that if λ = 0 then for 0 < < |λ| the image of the spectral projection PK (λ−, λ+) is finite dimensional. Without loss of generality we may assume that λ > 0. Proof. Let Kn be a sequence of finite rank operators such that If the image of the spectral projection PK (λ−, λ+) is infinite dimensional, we can find ψn in this image with Then

a contradiction for large enough n. Theorem 10.1.2. Suppose that K is the norm limit of finite rank operators. There exist orthonormal sets {ψj} and {ϕj} and positive numbers s j = s j (K) (called the singular values of K) such that

There are either finitely many s j (if K is of finite rank) or they converge to 0. We know that K ∗K is the norm limit of finite rank operators and is selfadjoint so we can apply the preceding theorem. We also know that Proof. Let be the non-zero eigenvalues of K∗K (arranged in decreasing order) and ϕj a corresponding orthonormal set. Then for any

Let

Then

To see that the ψj are orthonormal,

observe that

The formula for K∗ follows by taking adjoints. From K = j (·, ϕj)ψj we see that Kϕj = s jψj and from the formula for K∗ we see that K∗ψj = s jϕj. Hence

10.2

Compact Operators

Let K be a bounded operator on . Theorem 10.2.1. The following statements are equivalent: 1.

K is the norm limit of finite rank operators.

2.

If

3.

ψn →ψ weakly implies that Kψn → Kψ in norm.

4.

K is compact.

with An → A strongly then AnK → AK in norm.

We have already proved that 4 implies 1. 1 ⇒ 2 Replacing An by An − A we may assume that A = 0. By the uniform boundedness principle, there is some M such that is a sequence of finite rank operators with K j → K in norm, then so we may assume that K is of finite rank. Write K as the finite sum Then

2 ⇒ 3 Again replacing ψn by ψn −ψ we may assume that ψn →0 weakly. Choose any ϕ with and consider the rank one operators An ≔ (·,ψn)ϕ. Then An →0 strongly. Hence

But

3 ⇒ 4 If ψn is bounded, it has a weakly convergent subsequence. (Just choose an orthonomal basis ϕj and then a subsequence that all the (ψnk, ϕj) converge.) Then apply 3. to this subsequence. This completes the proof of the theorem. 10.3 10.3.1

Hilbert Schmidt Operators Hilbert Schmidt Integral Operators

Let (M,μ) be a measure space and K = K(x, y) ∈ L2(M × M,μ⊗μ). For ψ ∈ L2(M,μ) define K ψ by

So by Cauchy-Schwarz,

We see that K is a bounded operator on L2(M,μ).

Hilbert Schmidt operators are compact. Let ϕj be an orthonormal basis of L2(M,μ) so that ϕi (x)ϕ j (y) is an orthonormal basis of L2(M × M,μ⊗μ). We can expand K in terms of this orthonomal basis:

and hence

Since we see that K can be approximated in norm by finite rank operators and hence is compact. When is a compact operator on L2 Hilbert Schmidt? Let compact operator on with {s j} its singular values.

and K a

Proposition 10.3.1. K is Hilbert Schmidt if and only if

in which case K has an integral kernel with

Proof. We know that for orthonormal sets ϕj and ψj. Replacing this (possibly infinite) sum by a finite sum gives

as an approximating finite rank operator to K. Now K N has the integral kernel

Furthermore other. 10.3.2

. If one side converges, so does the

A More General Definition of Hilbert Schmidt Operator

Let us call a compact operator on a Hilbert space Hilbert Schmidt if In case we know that this coincides with our earlier definition. Since every Hilbert space is isomorphic to some L2(M,μ) we see that the Hilbert Schmidt operators together with the norm

form a Hilbert space isomorphic to L2(M × M,μ⊗μ). Here is another characterization of Hilbert Schmidt operators: Proposition 10.3.2. A compact operator K is Hilbert Schmidt if and only if there is an orthonormal basis {ζn} with

in which case this holds for any orthonormal basis. Furthermore

Proof. We have

All terms in the sums are positive so they converge or diverge together and this holds for any orthonormal basis.

10.3.3

An Important Example

Let = L2(ℝn) and let F denote the Fourier transform (extended to via Plancherel). For any function g we will let g(x) (bad notation) denote the operator of multiplication by g on (where defined). For a function f we will let f (p) (even worse notation) denote the operator

In other words, we take the Fourier transform then multiply by f (p) and then Fourier transform back. So, for example, the operator g(x) f (p) is given by

where denotes the inverse Fourier transform of f. If f and g are in L2(ℝn) then this kernel belongs to L2(ℝn ×ℝn) and so is Hilbert Schmidt. By symmetry, the operator f (p)g(x) is then also Hilbert Schmidt. Theorem 10.3.1. Let f and g be bounded (Borel measurable) functions which tend to 0 at∞. Then f (p)g(x) and g(x) f (p) are compact. Proof. By symmetry it is enough to consider g(x) f (p). Let fR be the function equal to f on the ball of radius R centered at the origin with a similar definition of gR. Then gR and fR belong to L2 and hence gR(x) fR(p) is Hilbert Schmidt and hence compact. We have

So gR(x) fR(p) approaches g f in norm and hence g(x) f (p) in norm. For example, suppose that H0 is the free Hamiltonian and we consider its resolvent at some negative real number, for example at −1. This operator is of the form f (p) where

Let V be a function which vanishes at∞. Then the operator V (x) f (p) is compact. This result will be of importance to us in conjunction with Weyl’s theorem on the essential spectrum. 10.4

Using Weyl’s Theorem for H 0 + V

We now use Proposition 9.4.1. A sequence such as the one in the proposition is called a singular Weyl sequence. We go back to Weyl’s theorem: Let A and B be self-adjoint operators. Let z be a point in the resolvent set of A and of B and let RA = RA(z) and RB = RB(z) be the corresponding resolvents. Theorem 10.4.1. If RA − R B is compact, then

Before proving the theorem observe the following identity involving the resolvent:

So if ψn is a singular Weyl sequence at λ then

Proof. Since the ψn converge weakly to 0 and RA − R B is compact, we know that (RA − R B)ψn converges strongly to zero and hence

In particular

But

so (after an innocuous renormalizing) the RB(z)ψn are a singular Weyl sequence for B. So σess (A) ⊂ σess (B). Interchanging the roles of A and B then proves the theorem. We now recall the second resolvent identity which says, for example, that if A and B are closed operators with D(B − A) ⊃ D(A) and z is in the resolvent set of both then

Definition 10.4.1. An operator K is relatively compact with respect to an operator A if KRA(z) is compact for some z in the resolvent set of A. Notice that from the first resolvent identity,

we see that if K RA(z) is compact for some z then K RA(z) is compact for all z in the resolvent set of A. So the condition is independent of z. From the second resolvent identity we see that if K is relatively compact with respect to a self-adjoint operator A then RA+K (z)− R A(z) is compact. Therefore we have the following theorem of Hermann Weyl: Theorem 10.4.2. If K is a self-adjoint operator which is relatively compact with respect to a self-adjoint operator A then σess(A + K) = σess(A). 10.4.1

Applications to Schrödinger Operators

In particular, in view of what we have proved above Proposition 10.4.1. If V is a bounded potential which tends to zero at infinity then

We now study the opposite situation—where the potential tends to infinity at infinity. We will show that under this circumstance H0 + V has no essential

spectrum. So its spectrum consists of eigenvalues of finite multiplicity. For this we will need a lemma. First observe that for all potentials W of compact support, the operator W(I + is compact. This is again because it is of the form f (p)g(x) where f and g are functions going to zero at infinity. Let V ≥ 0 and set H = H0 + V. Notice that H ≥ 0 as an operator, and in fact H ≥ H0. So for any

so

bounded and since (u, Hu) ≤ (u, (H + I)u) we see that

is bounded. Lemma 10.4.1. If V ≥ 0 is a non-negative potential and W is multiplication by a bounded function of compact support then W is H compact. Proof.

and we know that the product of the first two factors are compact and the product of the second two factors are bounded. So is compact. Multiply by the bounded operator on the right to conclude that W(I + H)−1 is compact. Theorem 10.4.3. If V (x)→∞ as x →∞ then σess(H0 + V) is empty. Proof. For any E write V − E = f − g where f ≥ 0 and g has compact support. By the lemma, g is H0 + f compact, so σess(H0 + f) = σess(H − E). Since f ≥ 0, we know that σess(H − E) ⊂ [0,∞) so σess(H) ⊂ [E,∞). Since this is true for all E we conclude that σess(H0 + V) is empty. 10.5

The Harmonic Oscillator

Consider the Schrödinger operator in n-dimensions:

In physics is a constant closely related to Planck’s constant. But we want to think of as a small parameter. By what we just proved, we know that if V →∞as x →∞then P(ħ) has discrete spectrum tending to +∞. 10.5.1

Weyl’s Law

Weyl’s law says that for any pair of real numbers a < b, the number of eigenvalues between a and b can be estimated by a certain volume in phase space:

Physicists know Weyl’s law as the “formula for the density of states.” I will not prove Weyl’s law, but will illustrate it in the case of the “harmonic oscillator,” where V is assumed to be a positive definite quadratic function of x. We built up the general case: This is taught in all elementary quantum mechanic courses. The operator P = P(1) is

We have

and hence e −x 2/2 is an eigenvector of P with eigenvalue 1. The remaining eigenvalues are found by the method of “spectrum generating algebras”: Define the creation operator

Here

and ix denotes the operator of multiplication by i x.We know that D is essentially self-adjoint. In fact, integration by parts shows that

for all smooth functions vanishing at infinity. Even more directly the operator of multiplication by ix is skew adjoint so we can write

in the formal sense. The operator A− is called the annihilation operator.

and

So we have proved

Notice that

so the first equation above shows again that

is an eigenvector of P with eigenvalue 1. Let v1 ≔ A+v0. Then

So v1 is an eigenvector of P with eigenvalue 3. Proceeding inductively, we see that if we define

then vn is an eigenvector of P with eigenvalue 2n +1. Also,

This allows us to conclude the (vn, vm) = 0 if m ≠ n. Indeed, we may suppose that m > n. Then since If n = 0 this is 0 since A −v0=0. If n > 0 then

By repeated use of this argument, we end up with a sum of expressions all being left multiples of A− and hence give 0 when applied to v0. We let

so that the un form an orthonormal set of eigenvectors. By construction, the vn, and hence the un, are polynomials of degree (at most) n times v0. So we have

and the Hn are called the Hermite polynomials of degree n. Since the un are linearly independent and of degree at most n, the coefficient of xn in Hn cannot vanish.

The un form a basis of L2(ℝ). To prove this, we must show that if g ∈ L2(ℝ) is orthogonal to all the un then g = 0. To see that this is so, if, (g, un) = 0 for all n, then (g, pe−x2/2) = 0 for all polynomials. Take p to be the n-th Taylor expansion of eix. These are all majorized by e|x| and e|x|e−x2/2 ∈ L2(ℝ). So from the Lebesgue dominated convergence theorem we see that (g, eix e−x2/2) = 0 which says that the Fourier transform of ge−x2/2 vanishes. This implies that ge−x2/2 ≡ 0. Since e−x2/2 does not vanish anywhere, this implies that g = 0. The next case, ħ = 1, n arbitrary.We may identify L2(ℝn) with the (completed) tensor product

where denotes the completed tensor product. Then the n-dimensional Schrödinger harmonic oscillator has the form

where P is the one-dimensional operator. So the tensor products of the u’s form an orthonormal basis of L2(ℝn) consisting of eigenvectors. Explicitly, let α = (α1, . . . ,αn) be a multi-index of non-negative integers and

Then the uα are eigenvectors of the operator

with eigenvalues

where

Furthermore the uα form an orthonormal basis of L2(ℝn). n = 1, ħ arbitrary. Consider the “rescaling operator”

This is a unitary operator on L2(ℝ) and on smooth functions we have

and

So

This shows that if we let

then the form an orthonormal basis of L2(ℝ) and are eigenvectors of eigenvalues The general case. We combine the previous two results and conclude that

are eigenvectors of

with eigenvalues

and the

form an orthonormal basis of L2(ℝn).

10.5.2

Verifying Weyl’s Law for the Harmonic Oscillator

with

In verifying Weyl’s law we may take a = 0 so we want to estimate

the number of lattice points in the simplex

This number is (up to lower order terms) the volume of this simplex. Also, up to lower order terms we can ignore the nħ in the numerator. Now the volume of the simplex is 1/n!× the volume of the cube. So

This gives the left-hand side of Weyl’s formula. As to the right-hand side,

is the volume of the ball of radius b in 2n-dimensional space which is πnbn/n!, as we recall below. This proves Weyl’s formula for the harmonic oscillator. Recall: the volume of spheres in ℝk. Let Ak−1 denote the volume of the k −1-dimensional unit sphere and Vk the volume of the k-dimensional unit ball, so

The

So

integral dx is evaluated as

the trick

The usual definition of the Gamma function is

If we set t = r 2 this becomes

So if we plug this back into the preceding formula we see that

Taking k = 2n this gives

and hence

10.6

Potentials in L2 ⊕ L∞ in ℝ3

Suppose that V = V1 + V2 with V1 ∈ L2(ℝ3) and V2 a bounded measurable function. For example, if V = 1/r, take where 1B1 is the indicator function of the unit ball, i.e., equal to 1 on the unit ball and zero outside it. Take V2 ≔ V − V1. Since 1/r is locally square integrable near the origin in three dimensions, and sup |V − V1|≤ 1 the potential 1/r has such a decomposition. We shall show that such a V satisfies

In fact, we will show that such a V satisfies (10.4) with a arbitrarily small: Since V2 is bounded we have for some M and this can be absorbed into the b in (10.4). So we must estimate The Fourier inversion formula (for ϕ ∈ D(H0)) tells us that

for any c > 0. So

The integral can be made as small as we like by choosing c large. Now

giving our desired estimate (10.4). If the potential V satisfies (10.4) and in addition satisfies V(x) →0 as x → ∞ then H0 + V has the same essential spectrum as H0. Indeed, the resolvent of H0 at −1 is Hilbert Schmidt and is compact, as proved above, so Weyl guarantees that H0 + V has the same essential spectrum as H0. 10.6.1

The Hydrogen Atom

We conclude that the essential spectrum of the hydrogen atom Hamiltonian is [0,∞). But are there any eigenvalues in this range? To show that there are none, I will apply the virial theorem to be stated and proved below: The quantum virial theorem. Let H = H0 + V and ψ be a state (with (ψ,ψ) = 1). Define the “kinetic energy” and the “potential energy” in the state ψ to be

Suppose that ψ is an eigenvector of H with eigenvalue λ and suppose that V is homogeneous of degree ρ, i.e., that

The virial theorem asserts that

Proof of (10.5): Proof. By definition, Tψ + Vψ = λ. Let ψa ≔ ψ(ax). Then

Differentiating the above equation with respect to a and setting a = 1 gives

But λ = Tψ + Vψ, proving (10.5). Application to the hydrogen Hamiltonian: all eigenvalues are negative: 1/r is homogeneous of degree −1 so ρ = −1 so (10.5) becomes Vψ = −2Tψ. But Tψ + Vψ = λ and Tψ > 0 so

The (negative) eigenvalues of can be calculated explicitly, and are calculated in most quantum mechanics books. This is usually done by writing down the partial differential equation which is formally equivalent to the eigenvalue problem, and solving it by separation of variables. There are two gaps in this argument: it is not clear a priori that every eigenfunction of the Hilbert space operator is sufficiently smooth to be a (classical) solution of the differential equation. Indeed, our treatment of the square well depended on the fact that in one dimension, elements of the domain of H0 were continuously differentiable, which need not be true in higher dimensions. Also, it is not clear that all solutions of the equation are obtained by separation of variables. A correct and complete treatment can be found in the paper “Atomic

Hamiltonians and the HVZ Theorem” by Kevin McLeod, available on the web. Later we will prove from an inequality due to Hardy that all eigenvalues are ≥ −Z2. There are no eigenvalues going off to−∞. In the physics literature this is known as the “stability of matter” for the case of the hydrogen atom—the electron cannot collapse into the nucleus. In general, the problem of the stability of matter is a major industry, including the work of eminent mathematicians such as Lieb. Some history of the virial theorem. The word virial derives from “vis,” the Latin word for “force” or “energy,” and was given its technical definition by Rudolf Clausius in 1870 who delivered the lecture “On a Mechanical Theorem Applicable to Heat” to the Association for Natural and Medical Sciences of the Lower Rhine, following a 20-year study of thermodynamics. The lecture stated that the mean vis viva of the system is equal to its virial, or that the average kinetic energy is equal to 1/2 the average potential energy. This theorem in classical mechanics was later utilized, popularized, generalized, and further developed by James Clerk Maxwell, Lord Rayleigh,Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux, and Eugene Parker. Fritz Zwicky was the first to use the virial theorem to deduce the existence of unseen matter, which is now called dark matter. As another example of the many applications of the classical version, the virial theorem has been used to derive the Chandrasekhar limit for the stability of white dwarf stars. Although originally derived for classical mechanics, the virial theorem in quantum mechanics, as stated above, derives from the basic work of Fock.

Chapter 11

Extending the Functional Analysis via Riesz 11.1

Integration According to Daniel

Daniel’s theory starts with a space L of bounded real valued functions on a set S which was closed under the operations ∧ and ∨, and a real valued function I on L which satisfies

From this, together with an important axiom of Stone, Daniel builds up a theory of integration. For details on this and on what follows, see “Abstract Harmonic Analysis” by Loomis. We will consider the case where L is the space of continuous functions of compact support on a Hausdorff space, more specifically on the real line. 11.2

A Riesz Representation Theorem for Measures

Theorem 11.2.1. Let F be a bounded linear function on L (with respect to the uniform norm). Then there is a complex valued measure μ on S such that

For a proof see Section 26.5 below, or see page 130 and page 619 of [21]. I will now show that we can use this theorem, together with the Riesz representation theorem on Hilbert space, to extend the map that we defined for f ∈ C0(ℝ) to all bounded Borel functions. I will temporarily use the notation O(f) (the operator corresponding to f) instead of f (A). Let f ∈ C0(ℝ). Fix vectors x, y in our Hilbert space . If f ∈ C0(ℝ) then

So the map

is a continuous linear function on the space of continuous functions of compact support bounded in norm by By the Riesz representation theorem for measures, this means that there is a unique complex valued bounded measure

on the Borel sets of ℝ such that

Since (O(f)x, y) depends linearly on x and anti-linearly on y, the uniqueness part of the Riesz representation theorem tells us that μx,y depends linearly on x and anti-linearly on y. So for each fixed Borel set U ⊂ ℝ, its measure μx,y(U) depends linearly on x and anti-linearly on y. From

we conclude that

and hence the uniqueness of the measure implies that

We can choose a sequence of fn ∈ C0(ℝ) which satisfy 0 ≤ fn ≤ 1 and which are monotone increasing to the constant function 1ℝ. It then follows from the monotone convergence theorem and

that and hence that μx,y defines a bounded measure on ℝ. So if f is now any bounded Borel function on ℝ, the integral

is well defined, and is bounded in absolute value by some constant times If we hold f and x fixed, this integral is a bounded anti-linear function of y, and hence by the Riesz representation theorem (for Hilbert space) there exists a w ∈ such that this integral is given by (w, y). The w in question depends linearly on f and on x because the integral does, and so: we can write w = O(f)x. 11.3

Verifying That

Is a Functional Calculus

We have defined a linear map O from bounded Borel functions on ℝ to bounded operators on such that

It is now routine to verify that the map satisfies the properties of of a functional calculus and so gives an extension of our functional calculus from continuous functions vanishing at infinity to bounded Borel functions. Here are some of the verifications: Verifying that : We have

Verifying multiplicativity which we know to be true for f, g ∈ C 0(ℝ). For f, g ∈ C0(ℝ) we have

Since this holds for all f ∈ C0(ℝ) (for fixed g, x, y) we conclude by the uniqueness of the measure that

Therefore, for any bounded Borel function f we have

This holds for all g ∈ C0(ℝ). By the uniqueness of the measure again, we conclude that μx,O(f)∗ y)= f μx,y hence

Chapter 12

Wintner’s Proof of the Spectral Theorem In this chapter I will present Wintner’s original (1929) proof of the spectral theorem which goes directly to Borel functions. It uses the Borel transform rather than the Fourier transform, and the Borel (Stieltjes) inversion formula (which I will explain) and the Helly selection principle. The exposition below mainly follows that in [25]. 12.1

Self-adjoint Operators and Herglotz Functions

Let A be a self-adjoint operator on a Hilbert space . Recall that every non-real z lies in the resolvent set of A and satisfies

The first resolvent identity applied to z and z∗ says that

and this equals

To not carry this minus sign around, let me set G = −R so

and

12.1.1 For any

Herglotz Functions define

Also, we know from the bound on the resolvent that

Now

Thus Fψ and z have the same imaginary part. In other words, Fψ is a holomorphic function defined on the complex plane with the real line removed, sends the half planes into themselves, and satisfies F (z∗) = F (z)∗. Such a function is called a Herglotz function. 12.1.2

The Borel Transform or the Stieltjes Transform

Let μ be a finite (real valued) non-negative Borel measure on ℝ. Its Borel transform B = B μ is defined as

Clearly B is holomorphic on ℂ \ supp(μ),

Also

so B is a Herglotz function. Conversely: Theorem 12.1.1. Let B be a Herglotz function satisfying

for z in the upper half plane for some constant M. Then there is a unique finite Borel measure μ on ℝ such that B is the Borel transform of μ. Furthermore, μ can be reconstructed from B via the Stieltjes inversion formula

The proof extends over the next few pages. This is a classical result. Proof of the existence of a μ. Let B(z) = v(z)+iw(z) and z = x+iy. Choose a contour Г consisting of the closed real interval of length 2R centered at x and shifted vertically by iand closed by a semi-circle in the upper half plane of radius R about x+i∈. So

Suppose that 0 < ∈ < y < R. Then z lies in the interior of Г and z∗ + 2i∈ lies in the exterior of Г. So by Cauchy’s theorem,

This equals

The integral over the semi-circle tends to zero as R→∞ yielding

The fraction, call it ϕ∈(λ), in this integral is real valued, and B = v+iw so taking the imaginary part of the above equation yields

Using the assumed bound

in the hypotheses of the theorem, we can let y→∞ to conclude that

For each fixed y > 0 we have | ϕ∈(y)− ϕ0(y)| < const. × ∈. So we pass to the limit and write

If we introduce the measures given by

(remember that w∈ is positive) we can write this as

Using the Helly selection principle. The bound μ∈(ℝ) ≤ M guarantees (by the Helly selection principle, see below) that we can select a subsequence μi of the μ∈ which converge “weakly” to a measure μ, meaning that for any continuous function f vanishing at ∞ we have

So we have found a measure μ such that

The Helly selection principle. The idea is that a bounded non-negative measure μ on ℝ is determined by the μ(−∞, λ), which is a bounded monotone increasing function of λ. So we have a family of such monotone increasing functions of λ and we can choose a subsequence, by Cantor’s diagonal method, to get a subsequence which converges at all rationals, and then extend to all reals by demanding (say) continuity on the left. This then gives the desired weakly converging subsequence. Now

So B(z) and

have the same imaginary part, hence, by Cauchy-Riemann, differ by a constant. From our assumption that we conclude that this constant is zero, and hence we have proved that there is (at least one) measure whose Borel transform is B. To show that μ is unique, it suffices to show that μ is given by the Stieltjes inversion formula as in the theorem:

The Stieltjes inversion formula. As we have already observed, . So the integral occurring on the right-hand side of

can be written (by Fubini’s theorem) as

We can do the inner integral to obtain

Since

we can apply the dominated convergence theorem to the outer integral to conclude the formula in the theorem. 12.2

The Measure μx.x

Applied to (G(z, A)x, x) we get a non-negative Borel measure, call it μx,x, whose Borel transform is (G(z, A)x, x). Going back to R = −G, we now know that for every x ∈ there is a measure μx,x such that

(Notice that I replaced λ−z by z −λ in the denominator to account for the change in sign in passing from G back to R.) 12.2.1 The Measure μx. y

By polarization, we obtain, for every pair x, y of elements of , a complex valued Borel measure μx,y on ℝ such that

The left-hand side is linear in x and anti-linear in y, and hence by the uniqueness in the theorem above, so is μx,y. 12.3

The Spectral Theorem via Wintner

For any bounded Borel function f consider the integral

This is a (bounded) anti-linear function of y, hence, by the Riesz representation theorem on Hilbert space, there is an ℓ = ℓ(f, x) ∈ such that

Similarly,

As ℓ depends linearly on x, there is a linear operator, call it O(f), such that

and

Also

so

To complete the proof of the spectral theorem, we need to show that the map

is multiplicative, i.e., that

This will be a consequence of the first resolvent identity! Lemma 12.3.1. Proof.

By the first resolvent identity this is

So μx,R(z∗,A)y and

have the same Borel transform and hence are equal.

Now

by the preceding lemma. So

Thus

proving that

12.4

Partitions of Unity or Resolutions of the Identity

Now that we have assigned a bounded operator to every bounded Borel function, we may define

for any Borel set U, where 1U is the indicator function of U, the function which is one on U and zero elsewhere. The following facts are immediate: 1. P(∅) = 0. 2. P(ℝ) = I, the identity. 3. P(U ∩ V) = P(U)P(V) and P(U)∗ = P(U). In particular, P(U) is a selfadjoint projection operator. 4. If U ∩ V = ∅ then P(U ∪ V) = P(U)+ P(V). 5. For each fixed x, y ∈ the set function Px.y : U → (P(U)x, y) is a complex valued measure. Such a P is called a resolution of the identity. It follows from the last item that for any fixed the map is an valued measure. Actually, given any resolution of the identity we can give a meaning to the integral

for any bounded Borel function f in the strong sense as follows: If s = Σ αi 1Ui is a simple function where ℝ = U1 ∪· · · ∪Un, and Ui ∩Uj = ∅ if i ≠ j and α1, . . . ,αn ∈ ℂ define

This is well defined on simple functions (is independent of the expression) and is multiplicative:

Also, since the P(U) are self-adjoint,

It is also clear that O is linear and

As a consequence, we get

so

If we choose i such that

and take x = P(Ui)y ≠ 0 we see that

provided we now take to denote the essential supremum of f which (we recall) means the following: It follows from the properties of a resolution of the identity that if Un is a sequence of Borel sets such that P(Un) = 0, then P(U) = 0 if U = ∪Un. So if f is any complex valued Borel function on ℝ, there will exist a largest open subset V ⊂ ℂ such that P(f−1(V)) = 0. We define the essential range of f to be the complement of V, say that f is essentially bounded if its essential range is compact, and then define its essential supremum to be the supremum of |λ| in the essential range of f. Furthermore, we identify two essentially bounded functions f and g if and call the corresponding space L∞(P). Every element of L∞(P) can be approximated in the norm by simple

functions, and hence the integral

is defined as the strong limit of the integrals of the corresponding simple functions. The map is linear, multiplicative, satisfies and (12.2) (with s replaced by f). 12.5

Stone’s Formula and the Stieltjes Inversion Formula

Stone’s formula gives an expression for the projection valued measure in terms of the resolvent. It fits right in with Wintner’s proof of the spectral theorem and the Stieltjes inversion formula. It says that for any real numbers a < b we have the strong limit

Although this formula cries out for a “complex variables” proof, and I will give one in Chapter 20, we can prove it just as we proved the Stieltjes inversion formula: Indeed, let

We have

The expression on the right is uniformly bounded, and approaches zero if x ∉ [a, b], approaches if x=a or x=b and approaches 1 if x ∈ (a, b). In short,

We may apply the dominated convergence theorem to f∈(A) to conclude Stone’s formula. Stone’s formula for f(A). As a consequence of the preceding we conclude that for any bounded Borel function f we have

Here is the front cover of Wintner’s 1929 book:

Please look at the sad biography of Helly at http://www-history.mcs.standrews.ac.uk/Mathematicians/Helly.html

Chapter 13

The L2 Version of a Spectral Theorem Our functional calculus version of the spectral theorem as discussed in the previous chapters is completely canonical. There is another version which is not canonical but extremely useful. It asserts that any self-adjoint operator on a separable Hilbert space is unitarily equivalent to a multiplication operator on L2(M, μ) of a suitable measure space. (For a rapid review of the relevant measure theory see the Appendix to [25] or the first two chapters of [21].) When using it we must be sure that our conclusions do not depend on the choices involved in the theorem. As mentioned, the (M, μ) is far from unique. We will find that we can take M to be the union of finite or countable copies of ℝ, each with its own measure. We will start with the case where there is a cyclic vector for A (see the next section for the definition) in which case we can take M to be ℝ with an appropriate measure. 13.1

The Cyclic Case

13.1.1 Cyclic Vectors A vector v ∈ is called cyclic for A if the linear combinations of all the vectors R(z, A)v as z ranges over all non-real complex numbers are dense in . Of course there might not be any cyclic vectors. But suppose that v is a cyclic vector. Consider the continuous linear function ℓ on C0(ℝ) given by

If f is real valued and non-negative, then In other words, ℓ is a non-negative continuous linear functional. As we have seen, Wintner’s approach to the spectral theorem or the Riesz representation theorem then says that there is a non-negative, finite, countably additive measure μ = μv,v on ℝ such that

In fact, from its definition, the total measure Let us consider C0(ℝ) as a (dense) subset of L2(ℝ,μ), and let (·, ·)2 denote the scalar product on this L2 space. Then for f, g ∈ C0(ℝ) we have

(where the last two scalar products are in ). This shows that the map

is an isometry from C0(ℝ) to the subspace of consisting of vectors of the form f (A)v. The space of vectors of the form f (A)v is dense in by our assumption of cyclicity (since already the linear combinations of vectors of the form rz(A), z ∉ ℝ are dense). 13.1.2

The Isometry U in the Cyclic Case

The space C0(ℝ) is dense in L2(ℝ). So the map above extends to a unitary map from L2(ℝ, μ) to whose inverse we will denote by is a unitary isomorphism such that

Now let f, g, h ∈ C0(ℝ) and set

Then

where, in this last term, the f denotes the operator of multiplication by f. In other words,

is the operator of multiplication by f on L2(ℝ, μ). In particular, U of the image of the operator f (A) is the image of multiplication by f in L2. Let us apply this last fact to the function f = rz, z ∉ ℝ, i.e.,

We know that the resolvent rz(A) maps onto the domain D(A), and that multiplication by rz, which is the resolvent of the operator of multiplication by x on L2, maps L2 to the domain of the operator of multiplication by x. This latter domain is the set of k ∈ L2 such that xk(x) ∈ L2. Now (zI − A)rz(A) = I, so

Applied to U−1g, g ∈ L2(R, μ) this gives

So

and multiplying by U gives

If y ∉ supp(μ) then multiplication by ry is bounded on L2(ℝ, μ) and conversely.

So the support of μ is exactly the spectrum of A. 13.1.3 The General Case For a general separable Hilbert space with a self-adjoint operator A we can decompose into a direct sum of Hilbert spaces invariant under A each of which has a cyclic vector. Here is a sketch of how this goes. Start with a countable dense subset {x1, x2, . . .} of . Let be the cyclic subspace generated by x1, i.e., is the smallest (closed) cyclic subspace containing x1. Let m(1) be the smallest integer such that Let ym(1) be the component of xm(1) orthogonal to , and let be the cyclic subspace generated by ym(1). Proceeding inductively, suppose that we have constructed the cyclic subspaces and let m(n) be the smallest integer for which xm(n) does not belong to the (Hilbert space direct sum) . Let ym(n) be the component of xm(n) orthogonal to this direct sum and let by the cyclic subspace generated by y(m). At each stage of the induction there are two possibilities: If no m(n) exists, then is the finite direct sum If the induction continues indefinitely, then the closure of the infinite Hilbert space direct sum · contains all the xi and so coincides with By construction, each of the spaces i is invariant under all the R(z, A) so we can apply the results of the cyclic case to each of the i. Let us choose the cyclic vector vi ∈ i to have norm 2−n so that the total measure of ℝ under the corresponding measure μi is 2−2n. Recall that S denotes the spectrum of A and each of the measures μi is supported on S. So we put a measure μ on S×ℕ so that the restriction of μ to S ×{n} is μn. Then combine the Un given above in the obvious way. We obtain the following theorem: Theorem 13.1.1. Let A be a self-adjoint operator on a separable Hilbert space and let S = Spec(A). There exists a finite measure μ on S×ℕ and a unitary isomorphism

such that U AU−1 is multiplication by the function a(s, n) = s. More precisely, U

takes the domain of A to the set of functions h ∈ L2 such that ah ∈ L2 and for all such functions h we have

For any f ∈ C0(ℝ) we have

In particular, if supp(f) ∩ S = ∅ then f (A) = 0. This is our L2 version of the spectral theorem. Its proof depended on many choices so it is not canonical. Nevertheless, the rest of this chapter will be devoted to giving applications of this theorem which do not depend on our choices. 13.2

Fractional Powers of a Non-negative Operator

13.2.1

Non-negative Operators

Recall: Definition 13.2.1. A symmetric operator H with domain D is non-negative if

More generally, we say that for all f ∈ D. The following theorem is an immediate consequence of our L2 version of the spectral theorem and the spectral properties of multiplication operators: Theorem 13.2.1. The following are equivalent for a self-adjoint operator H: • • •

H ≥ c. The spectrum of H is contained in [c,∞). In a spectral L2 representation the function except on a set of measure zero.

(is nonnegative

The case c = 0 states that a non-negative self-adjoint operator H is unitarily equivalent to a non-negative multiplication operator h on an L2 space. For any λ ∈ ℝ we would like to define the operator H λ so that it is unitarily

equivalent to the multiplication operator hλ. Since the L2 version of the spectral theorem is not canonical, we must check that this is well defined. H λ is well defined. For this, consider the function f on the real line given by

f is a Borel function (actually continuous) so our functional calculus version of the spectral theorem tells us that there is a unique self-adjoint operator G such that G = f (H). In any L2 spectral representation G corresponds to the multiplication operator of the form (hλ +1)−1 and so there is a (generally unbounded) canonical self-adjoint operator K such that (K +1)−1 = f (H), and the uniqueness of the map tells us that K is canonically defined, so H λ is well defined. 13.2.2

The Case 0 < λ < 1

Theorem 13.2.2. If H is a non-negative self-adjoint operator and 0 < λ < 1, then

in which case

Proof. It is enough to prove this in an L2 spectral representation in case the theorem becomes the obvious statement that if and only if both

13.2.3 The Case the Quadratic Form Associated with a Non-negative Self-adjoint Operator Let H be a non-negative self-adjoint operator. For f, g ∈ Dom

define

Proposition 13.2.1. f ∈ Dom(H) if and only if f such that

for all g ∈ Dom

and there exists a k ∈

If this happens then Hf = k.

Proof. The equation in the proposition asserts that is self-adjoint, the proposition is the special preceding theorem. 13.3

with of the

The Lax-Milgram Theorems

Over the next few pages we follow the treatment in Helffer [6]. Let be a Hilbert space, and a a continuous sesquilinear form on meaning that there is a constant C such that

(Here the norms are those of .) Since a is anti-linear and continuous in v we know by the Riesz representation theorem for Hilbert spaces that there is a bounded linear map A: such that

Definition 13.3.1. a is -elliptic if there exists an α > 0 such that

Theorem 13.3.1 (Lax-Milgram-I). If a is operator A is an isomorphism.

elliptic, then the associated

Proof. From a(u, v) = (Au, v) the ellipticity and Cauchy-Schwarz we conclude that

So

Proposition. The image of A is dense. Indeed if u is orthogonal to the image of A, then = a(u, u) implying that u = 0 by the ellipticity. Proposition. The image of A is closed: Let {vn} be a Cauchy sequence in the image of A, and {un} a sequence such that Aun = vn. From (13.5) we conclude that the un converge to some w and the continuity says that vn → v = Aw. The above two facts show that A is surjective and (13.5) implies that A−1 is bounded with inverse α−1, completing the proof of Lax-Milgram-I. 13.3.1

Gelfand’s Rigged Hilbert Spaces

Let and be two Hilbert spaces together with a continuous injection of into which we write sloppily as

with the continuity of the injection meaning, as usual, that there is some constant C > 0 such that for any

We also assume that

The injection of on given by

into the dual space

: If h ∈ , then the linear function

is continuous. So gives a map of , the dual space of . This map is injective, for the density of in tells us that 0 = ℓh(u) = (u, h) for all implies that h = 0. So we have

Now let a be a continuous (with respect to elliptic sesquilinear form as above. We will associate to a an unbounded operator S on as follows: We define the domain D(S) of S to consist of all such that the map is

continuous on for the topology induced by Since we are assuming that is dense in this extends to a unique continuous anti-linear function on and hence by the Riesz representation theorem applied to we conclude that we get an Su ∈ such that

Theorem 13.3.2 (Lax-Milgram-II). S is a bijective map from D(S) onto with S−1 a bounded operator on . Furthermore, D(S) is dense in We will break the proof into a number of steps: S is injective. Proof. By the continuity of the injection for u ∈ D(S),

and the ellipticity of a we have,

So

Notice that (13.9) implies that S−1 is bounded on its image. S is surjective. Proof. Let Then is a continuous anti-linear function on so by the Riesz representation theorem for there is a such that

Let A be the (invertible) linear operator on given by the Lax-Milgram theorem. Let u = A−1w so

Therefore u ∈ D(S) and Su = , proving that bijective. D(S) is dense in

We must show that

is surjective, and hence implies that h = 0.

Proof. By the surjectivity of S we can find a v ∈ D(S) such that Sv = h. So (Sv, u) for all In particular by our ellipticity assumption, so v = 0. 13.3.2

The Hermitian Case—Lax-Milgram-III

We now specialize to the case that a is Hermitian, i.e., that

in addition to the previous assumptions. Then Theorem 13.3.3. 1. S is closed, 2. S = S∗, 3. D(S) is dense in Notice that 2) ⇒ 1) since adjoints are closed. Proof that S = S∗. Proof. Hermiticity implies that for u, v ∈ D(S) we have the operator S on is symmetric. In particular,

i.e., that

We must show that D(S) = D(S∗). So let v ∈ D(S∗). The surjectivity of S tells us that there is a w ∈ D(S) such that

So for all u ∈ D(S) we have

The surjectivity of S now implies that v = w ∈ D(S) and Sv = S∗v. Proof that D(S) is dense in .

Proof. Suppose that that Af = h, where u ∈ D(S) we have

is orthogonal to D(S) in the metric. Choose such is the isomorphism given by Lax-Milgram. So for all

But a(u, f) = (Su, f) and the surjectivity of S implies that f = 0 and hence h = 0. 13.4

Semi-bounded Operators and the So-called Friedrichs Extension

13.4.1

Semi-bounded Operators

Definition 13.4.1. Let T0 be a symmetric unbounded operator on a Hilbert space with domain D. We say that T0 is semi-bounded (from below) if there exists a constant C such that

13.4.2

The Friedrichs Extension

Theorem 13.4.1. A symmetric semi-bounded operator T0 whose domain D = D(T0) is dense in has (at least one) self-adjoint extension. Replacing T0 by T0 + cI for a suitable c we may assume that the operator T0 satisfies

We have the Hermitian form a0 defined on D × D by

and (13.11) says that

The space Consider the norm p0 on D given by

and let denote the completion of D relative to this norm. inherits a scalar product given by

where {un} and {vn} are Cauchy sequences in D for p0 tending, respectively, to u and v in The inequality (13.12) implies that

so the injection of into is continuous, and since We are now in the context

and D is dense in so is

of Lax-Milgram-III with

so we get an unbounded self-adjoint operator S on extending T0 whose domain D(S) is contained in . This completes the proof of the existence of the Friedrichs extension. This theorem, attributed to Friedrichs (1932), is found in Wintner’s book of 1929. 13.5

Hardy’s Inequality and the Hydrogen Atom

The hydrogen Hamiltonian. Up to various parameters (which I will absorb into the choice of units so as not to complicate the appearance of the formulas), I remind you that this is the operator (initially defined on given by the operator

where H0 is the “free Hamiltonian” that

We will show

so that SZ is semi-bounded and hence has a Friedrichs extension. The proof of (13.13) hinges on an inequality that goes back to G.H. Hardy. 13.5.1

Hardy’s Inequality

This says that for u ∈ C0(ℝ3) we have

For the proof of this inequality, observe that

so that

I will now massage the right-hand side of (13.16). Integrating by parts. We have (since u is assumed to have compact support)

so summing over i gives

Substituting this into (13.16) gives Hardy’s inequality. Using Cauchy-Schwarz and Hardy. By Cauchy-Schwarz and Hardy (and r = we have

We now use our old trick

Taking

gives

to conclude that

Chapter14

Rayleigh-Ritz Instead of an exact calculation, I will discuss an extremely important method of approximate calculation known generally as the “Rayleigh-Ritz” method. I will follow the treatment in the book [4] by Davies. Lord Rayleigh was one of the most famous physicists of the 19th and 20th centuries, winning the Nobel prize for his discovery of argon, for example. What is his relation to the “Rayleigh-Ritz” method? Here is the abstract of a paper “The Historical Bases of the Rayleigh and Ritz Methods” by A.W. Leissa, which appeared in the Journal of Sound and Vibration, Vol. 287, pp. 961–978: “Rayleigh’s classical book Theory of Sound was first published in 1877. In it are many examples of calculating fundamental natural frequencies of free vibration of continuum systems (strings, bars, beams, membranes, plates) by assuming the mode shape, and setting the maximum values of potential and kinetic energy in a cycle of motion equal to each other . . . . This procedure is well known as Rayleigh’s Method. In 1908, Ritz laid out his famous method for determining frequencies and mode shapes, choosing multiple admissible displacement functions, and minimizing a functional involving both potential and kinetic energies. He then demonstrated it in detail in 1909 for the completely free square plate. In 1911, Rayleigh wrote a paper congratulating Ritz on his work, but stating that he himself had used Ritz’s method in many places in his book and in another publication. Subsequently, hundreds of research articles and many books have appeared which use the method, some calling it the Ritz method and others the Rayleigh-Ritz method. The present article examines the method in detail, as Ritz presented it, and as Rayleigh claimed to have used it. It concludes that, although Rayleigh did solve a few problems which involved minimization of a frequency, these solutions were not by the straightforward,

direct method presented by Ritz and used subsequently by others. Therefore, Rayleigh’s name should not be attached to the method.” Who was Ritz? Walther Ritz (February 22, 1878–July 7, 1909) was a Swiss theoretical physicist. His father, Raphael Ritz, a native of Valais, was a wellknown landscape and interior scenes artist. His mother was the daughter of the engineer Noerdlinger of Tubingen. Ritz studied in Zurich and Göttingen. He is most famous for his work with Johannes Rydberg on the Rydberg-Ritz combination principle. Ritz is also known for the variational method named after him, the Ritz method. Ritz died in 1909, at the age of 31. According to Forman’s “Dictionary of Scientific Biography,” Ritz contracted tuberculosis in 1900, which led to his death. According to Ritz’s collected works the disease was pleurisy. 14.1

The Rayleigh-Ritz Method

Let H be a non-negative self-adjoint operator on a Hilbert space For any finite dimensional subspace L of with L ⊂ D ≔ Dom(H) define

Define

The λn are an increasing family of numbers. We shall show that they constitute that part of the discrete spectrum of H which lies below the essential spectrum. Here is the relevant theorem: Theorem 14.1.1. Let H be a non-negative self-adjoint operator on a Hilbert space . Define the numbers λn = λn(H) by (14.1). Then one of the following three alternatives holds: 1. H has empty essential spectrum. Then the λn →∞ and coincide with the eigenvalues of H repeated according to multiplicity and listed in increasing order or else is finite dimensional and the λn coincide with the eigenvalues of H repeated according to multiplicity and listed in increasing order. 2. There exists an a < ∞ such that λn < a for all n and limn→∞ λn = a. In this

case a is the smallest number in the essential spectrum of H and σ(H) ∩ [0, a) consists of the λn which are eigenvalues of H repeated according to multiplicity and listed in increasing order. 3. There exists an a < ∞and an N such that λn < a for n ≤ N and λm = a for all m > N. Then a is the smallest number in the essential spectrum of H and σ(H) ∩[0, a) consists of λ1, . . . ,λN which are eigenvalues of H repeated according to multiplicity and listed in increasing order. Let b be the smallest point in the essential spectrum of H (so b=∞ in case 1). So H has only isolated eigenvalues of finite multiplicity in [0, b) and these constitute the entire spectrum of H in this interval. Let {fk} be an orthonormal set of eigenvectors corresponding to these eigenvalues μk listed with multiplicity in increasing order. We want to show that

Proof. We first show that

Let Mn denote the space spanned by the fk, k ≤ n, and let f ∈ Mn. Then

so

so

proving that λn ≤ μn. In the other direction, let L be an n-dimensional subspace of Dom(H) and let P denote the orthogonal projection of onto Mn−1 so

The image of P restricted to L has dimension at most n −1 while L has dimension n. So there must be some f ∈ L with Pf = 0. By the multiplicative version of the spectral theorem, the function corresponding to f is supported in the set where h ≥ μn. Hence (H f, f) ≥ μn║ f ║2 so

We have proved (14.2). There are now three cases to consider: If b = +∞ (i.e., the essential spectrum of H is empty) then μn = λn can have no finite accumulation point so we are in case 1). If there are infinitely many μn in [0, b), b < ∞ they must have a finite accumulation point a ≤ b, and a is in the essential spectrum. Then we must have a = b and we are in case 2). The remaining possibility is that there are only finitely many μ1, . . . ,μn < b. Then for k ≤ M we have λk = μk as above and λm ≥ b for m > M. Since b ∈ σess(H), the space K ≔ P(b − ∈, b + ∈) is infinite dimensional for all> 0. Let {f1, f2, . . .} be an orthonormal basis of K and let L be the space spanned by the first m of these basis elements. By the spectral theorem, for any f ∈ L. So for all m and > 0 we have λm ≤ b+∉. We are in case 3). In applications (say to chemistry) we deal with self-adjoint operators which are bounded from below, rather than being non-negative. But this requires just a trivial shift in stating and applying the preceding theorem. In some of these applications the bottom of the essential spectrum is at 0, and we are interested in the lowest eigenvalue λ1 which is negative. The hydrogen atom. Here is a famous application of the method: The Hamiltonian for the hydrogen atom is

where κ0 ≔ 4π∈0 with 0 the permittivity of free space, and m and e are the mass and charge of the electron. We want to minimize

to find the lowest eigenvalue. Suppose we try functions of r alone, ψ = ψ(r). Then

More specifically, try

Then

In doing the integrals (with respect to r2 sin θdrdθdϕ) observe that

so

The minimum over c is achieved at

giving the value

It turns out that this is the lowest eigenvalue of H. This is due to our choice of the exponential function as trial function. Had we chosen a Gaussian as trial function, we would be off by around 15%. The Bohr radius. According to Bohr’s 1913 theory of the hydrogen atom, the electron moves around the nucleus in a circle of radius

We see that the value of c that we found for the minimum is 1/a0. So if we choose our units of length so that a0 = 1 (the so-called “atomic units”) then c = 1 and

In fact, in these units the function e−r is an eigenvector of H with the lowest eigenvalue λ1. 14.1.1

The Hydrogen Molecule

In 1929, Dirac made the outrageous statement “The underlying physical laws necessary for the mathematical theory of a large part of physics and all of chemistry are thus completely known.” Let’s examine how Rayleigh-Ritz enters into the “theory”’ of the hydrogen molecule. A starting approximation is to assume that each nucleus is a point particle. That is, we ignore the fact that the nucleus is composed of protons and neutrons, and that these are made of quarks, etc. The Born-Oppenheimer approximation (1927). This approximation is to treat the nuclei as fixed (at a distance R from one another) and try to solve the Schrödinger operator for the pair of electrons, in particular to find its lowest eigenvalue. Add to this lowest eigenvalue (the so-called “electronic energy”) the energy of repulsion between the nuclei to obtain a function E (R). Then search

for the R which minimizes E (R). The Hamiltonian of the electrons, expressed in units of

has the form

Here is a typical graph of E (R) taken from a standard text [3]. The zero of energy is chosen at infinite separation. E (R) decreases below zero as R decreases, with a minimum at R E, called the equilibrium distance. The depth De of the minimum of this curve is the “electronic dissociation energy.” The actual dissociation energy Do is a bit less due to the energy of vibration.

14.1.2

The Heitler-London “Theory” (1927)

If the nuclei are very far apart, we expect the ground state to be given by the tensor product of the ground states of the individual hydrogen atoms, there being no interaction between them. If we call the nuclei a and b, and the corresponding ground states ϕa and ϕb, then we might expect the combined ground state to be of the form ϕa ⊗ ϕb. So we might expect that the combined ground state be of the form ϕa(1) ⊗ ϕb(2) where the notation ϕa(1) implies that it is electron 1 which is near nucleus a. But how do we know that it is electron 1 and not electron 2 which is around nucleus a? So Heitler and London suggested that we apply Rayleigh-Ritz using the twodimensional subspace of the tensor product spanned by ϕa(1)⊗ϕb(2) and

ϕa(2)⊗ϕb(1). On the basis of symmetry, we can conclude that the extrema of the quadratic form restricted to this two-dimensional subspace must be at the combinations

The value of the quadratic form evaluated on Φ+ or Φ− is reduced to an integral. These integrals were difficult to evaluate, and it took another year for their evaluation. But then we obtained two curves, one corresponding to Φ+ which was of the form of the figure, and one corresponding to Φ− which had no minimum. The minimum for Φ+ indicated an equilibrium bond length of 87 pm, compared with the observed experimental value of 74 pm, and a disassociation energy of 3.14 ev compared with the experimental value 4.75 ev. Not bad! In 1916, the chemist G.N. Lewis proposed a “theory” of the “covalent bond” as being due to the “sharing of two electrons” between the two atoms. HeitlerLondon’s use of Rayleigh-Ritz might be considered as a derivation of the Lewis theory. This is the “valence bond” theory of Heitler, London, Pauling, and Slater. Other “theories” are currently in use. 14.1.3

Variations on the Variational Formula

Instead of (14.1) we can determine the λn as follows. We define λ1 as before:

Suppose that f1 is an f which attains this minimum. We then know that f1 is an eigenvector of H with eigenvalue λ1. Now define

This λ2 coincides with the λ2 given by (14.1) and an f2 which achieves the minimum is an eigenvector of H with eigenvalue λ2. Proceeding this way, after finding the first n eigenvalues λ1, . . . ,λn and corresponding eigenvectors f1, . . . , fn, we define

This gives the same λk as (14.1). Variations on the condition L ⊂ Dom(H). In some applications, the condition L ⊂ Dom(H) is unduly restrictive, especially when we want to compare eigenvalues of different self-adjoint operators. In these applications, one can frequently find a common core D for the quadratic forms Q associated to the operators. That is,

and D is dense in Dom

for the metric ║·║1 given by

where

Theorem 14.1.2. Define

Then

Proof. Since D ⊂ Dom

the condition L ⊂ D implies

Conversely, given ∈ > 0 let L ⊂ Dom

so

be such that L is n-dimensional and

Restricting Q to L × L, we can find an orthonormal basis f1, . . . , fn of L

such that

We can then find gi ∈ D such that

for all i = 1, . . . , n. This means that

and

and the constants cn and depend only on n. Let Ľ be the space spanned by the gi. Then Ľ is an n-dimensional subspace of D and

satisfies

where depends only on n. Letting ∈→0 shows that To complete the proof of the theorem, it suffices to show that Dom(H) is a core for Dom This follows from the spectral theorem: The domain of H is unitarily equivalent to the space of all f such that

where h(s, n) = s. This is clearly dense in the space of f for which

since h is non-negative and finite almost everywhere. 14.1.4

The Secular Equation

The definition (14.1) makes sense in a real finite dimensional vector space. If Q is a real quadratic form on a finite dimensional real Hilbert space V, then we can write Q(f) = (H f, f) where H is a self-adjoint (= symmetric) operator, and then find an orthonormal basis according to (14.1). In terms of such a basis f1, . . . , fn, we have

If we consider the problem of finding an extreme point of Q (f) subject to the constraint that (f, f) = 1, this becomes (by Lagrange multipliers), the problem of finding λ and f such that

In terms of the coordinates (r1, . . . , rn) we have

So the only possible values of λ are λ = μi for some i and the corresponding f is given by r j = 0, j ≠ i and ri ≠ 0. This is a watered down version version of Rayleigh-Ritz. In applications, we are frequently given a basis of V which is not orthonormal. Thus (in terms of the given basis)

where

The problem of finding an extreme point of Q (f) subject to the constraint S (f) = 1 becomes that of finding λ and r = (r1, . . . , rn) such that

which is known as the secular equation due to its previous use in astronomy to determine the periods of orbits. 14.2

Back to Chemistry: Valence

The minimum eigenvalue λ1 is determined according to (14.1) by

Unless we have a clever way of computing λ1 by some other means, minimizing the expression on the right over all of is a hopeless task. What is done in practice is to choose a finite dimensional subspace and apply the above minimization over all ψ in that subspace (and similarly to apply (14.1) to subspaces of that subspace for the higher eigenvalues). The hope is that this yields good approximations to the true eigenvalues. We saw how this worked in the Heitler-London theory. If M is a finite dimensional subspace of , and P denotes projection onto M, then applying (14.1) to subspaces of M amounts to finding the eigenvalues of PHP, which is an algebraic problem as we have seen. A chemical theory (when H is the Schrödinger operator) then amounts to cleverly choosing such a subspace. 14.2.1

Two-dimensional Examples

Consider the case where M is two-dimensional with a basis ψ1 and ψ2. The idea is that we have some grounds for believing that the true eigenfunction has characteristics typical of these two elements and is likely to be some linear combination of them. If we set

and

then if these quantities are real we can apply the secular equation

to determine λ. Suppose that S11 = S22 = 1, i.e., that ψ1 and ψ2 are separately normalized. Also assume that ψ1 and ψ2 are linearly independent. Let

This β is sometimes called the “overlap integral” since if our Hilbert space is L2(ℝ3) then Now

is the guess that we would make for the lowest eigenvalue (= the lowest “energy level”) if we took L to be the one-dimensional space spanned by ψ1. So let us call this value E1. So E1 ≔ H11 and similarly define E2 = H22. The secular equation becomes

If we define F(λ) ≔ (λ− E1)(λ− E2)−(H12−λβ)2 then F is positive for large values of |λ| since |β| < 1 by Cauchy-Schwarz. F(λ) is non-positive at λ = E1 or E2 and in fact generically will be strictly negative at these points. So the lower solution of the secular equations will generically lie strictly below min(E1, E2) and the upper

solution will generically lie strictly above max(E1, E2). This is known as the no crossing rule and is of great importance in chemistry. 14.2.2

The Hückel Theory of Hydrocarbons

In this theory, the space M is the n-dimensional space where each carbon atom contributes one electron. (The other electrons are occupied with the hydrogen atoms.) It is assumed that the S in the secular equation is the identity matrix. This amounts to the assumption that the basis given by the electrons associated with each carbon atom is an orthonormal basis. It is also assumed that (H f, f) = α is the same for each basis element. In a crude sense this measures the electronattracting power of each carbon atom and hence is assumed to be the same for all basis elements. If (H fr, fs) ≠ 0, the atoms r and s are said to be “bonded.” It is assumed that only “nearest neighbor” atoms are bonded, in which case it is assumed that (H fr, fs) = β is independent of r and s. So PH P has the form

where A is the adjacency matrix of the graph whose vertices correspond to the carbon atoms and whose edges correspond to the bonded pairs of atoms. If we set

then finding the energy levels is the same as finding the eigenvalues x of the adjacency matrix A. In particular, this is so if we assume that the values of α and β are independent of the particular molecule.

Chapter 15

Some One-dimensional Quantum Mechanics 15.1

Introduction

In this chapter we will study the Hamiltonian H = H0 + v in one dimension where v grows at infinity so that H has a purely discrete spectrum. In fact, what we will say applies more generally to the situation where we study the eigenvalues below the essential spectrum. We will show that if the eigenvectors are arranged in increasing order then the k-th eigenfunction has exactly k zeros. The fact that it has at least k zeros is a 19th century result, going back to the work of Sturm (1803–1855). The fact that it has no more than k zeros is a nice application of the theory of quadratic forms and Rayleigh-Ritz. We then sketchily describe analogous results in higher dimensions. Everything in this chapter is taken from [22]. 15.2 15.2.1

The Sturm Comparison Theorem and Its Consequences The Sturm Oscillation Theorem

Let y1, y2 be non-zero real solutions of

Theorem 15.2.1. If v1 ≥ v2 on an interval [a, b] with y1(a) = y1(b) = 0, then there exists x0 ∈ [a, b] such that y2(x0) = 0. In words: between any two zeros of y1 there is a zero of y2. For the proof, wemay assume that there are no zeros of y1 in the open interval (a, b) (otherwise pass to a smaller interval) and since −y1 is just as good a solution as y1 we may assume that y1 > 0 on (a, b), and hence by the uniqueness theorem for ordinary differential equations that and Proof. Suppose that there are no zeros of y2 in [a, b] and so without loss of generality we may assume that y2 > 0 on [a, b]. Multiplying the equation for y1 by y2 and the equation for y2 by y1 and subtracting, we get

so

which is a contradiction. A stronger hypothesis and conclusion. Suppose that we have v1 ≥ v2 on [a, b] and the strict inequality v1 > v2 on a set of positive measure. Then the hypothesis that y1 y2 > 0 on (a, b) implies that and hence that which contradicts the fact that and y2(a) ≥ 0, y2(b) ≥ 0. So under this stronger hypothesis we can conclude that y2 has a zero in the open interval (a, b). A corollary. If we take v2 ≡ 0 the equation has as a solution y2 ≡ 1 which has no zeros on any interval. So we can conclude: Corollary 15.2.1. If v ≥ 0 on [a, b] then a non-vanishing solution of −y″ + vy = 0 has at most one zero on [a, b].

15.2.2

A Dichotomy

Suppose that v(x) ≥ ∈ > 0 for x ≥ a. By the preceding corollary, we can suppose that the solution of

is > 0 for x > a′ ≥ a, so y″′ > 0 and hence y' is non-decreasing in this range. Replace a' by a. So there are two possibilities: 1.

there exist some α > a such that y'(α) > 0, or

2.

y'(x) ≤ 0 ∀ x > a.

In the first case we have y'(x) ≥ y'(α) > 0 for all x > α so y(x) → +∞ as x →∞. In the second case, y is steadily decreasing and hence tends to some limit γ, and hence y' → 0. If γ > 0 then y" ≥ γ and hence y'(x) ≥ y'(α)+ (x −α)∈γ so y' → ∞ contradicting 2). So γ = 0. We have proved: Theorem 15.2.2. Suppose that v(x) ≥ ∈ > 0 for x ≥ a and y is a solution of −y″ + vy = 0 with y(x) > 0 for x > a. Then either

Solutions with the first alternative exist as we can choose initial conditions y(a) > 0, y'(a) > 0. So solutions satisfying the second alternative are unique to within a factor. We now prove the existence of such decaying solutions: The existence of decaying solutions, 1. Consider the set of solutions with the initial conditions

and let

t ∈ M+ if there exists x0 > a with yt(x0) > 0 and By continuous dependence of solutions on initial conditions, if this holds for t0, it holds for all nearby t so M+ is open. Similarly, M− is open. Clearly M+ ∩ M − = ∅, and since ℝ is connected, we cannot have M+ ∪ M− = ℝ if M− = ∅ since we know that M+ ≠ ∅. So to prove the existence of decaying solutions, it is enough to prove that M − ≠ ∅. So we must show that for some value of t we have an x0 > a with yt (x0) < 0. The existence of decaying solutions, 2. M − ≠ ∅: We have, for x > a,

Now yt(x) < y0(x) for all t < 0 and all x > a because yt (x)− y0(x) can have at most one zero. So maxt ≤ 0,ξ∈[a,a+1][v(x)yt (x)] ≤ C for some constant C. So taking x = a +1 in (∗) we get

15.2.3

Comparison of Two Equations

Theorem 15.2.3. Let corresponding equations:

and let y1, y2 be solutions of the

If yi (x) → ∞ as x → ∞, i = 1, 2, then ∃ c > 0 such that

If y1, y2 are positive decaying solutions then ∃ c > 0 such that

Proof. (1) Recall from the proof of the previous theorem that increasing solutions have the property that there is some α > a with . Choose c > 0 such that

Our choice of c says that z(α) > 0 and z'(α) > 0. We want to show that

If not, ∃b > α with z(b) = 0, z'(b) ≤ 0. Apply the mean value theorem to z', to find a ξ ∈ [α, b] with z˝ (ξ) < 0 contradicting the above. This proves the first assertion in the theorem. Proof. (2) Set z = y1 − cy2 where c is such that z(a) < 0. Then . We want to prove that z(x) ≤ 0 for all x > a. If not, there exists an α > a with z(α) > 0, z'(α) > 0. But then we can argue as before that

Indeed, if not we can find a b > α such that z(b) = 0, z'(b) ≤ 0 and applying the mean value theorem to z' to conclude the existence of ξ ∈ [α, b] with z"(ξ) < 0 contradicting z"−v2z ≥ 0. But the conclusion that z(x) ≥ z(α) > 0 for all x > α contradicts the assumption that y1 and y2 →0. Corollary 15.2.2. Suppose that v(x) → ∞ as x → ∞. Then any solution of −y" + vy = 0 either grows faster than ekx at ∞ or dies faster than e−kx for any k > 0. This follows by comparing the equation with −y" + k2y = 0. Over the next few pages we assume that v(x) → ∞ as |x| → ∞ and that y is a non-vanishing solution of

Zeros of solutions. Since there is an N such that v(x) ≥ 0 for |x| ≥ N we know

that v can have at most one zero on [N,∞) and on (−∞,−N]. The zeros have to be discrete by the uniqueness theorem of differential equations, hence Proposition 15.2.1. y has at most a finite number of zeros (possibly none at all). 15.2.4

Application to Eigenvectors

We want to apply this to the eigenvalues of

i.e., to solutions of

which vanish (sufficiently fast) at ±∞. Let λ1 > λ2 be eigenvalues of H. So v −λ2 > v−λ1 so between any two zeros of an eigenfunction with eigenvalue λ1 there is a zero of any eigenfunction with eigenvalue λ2, by our comparison theorem. Let y1 be an eigenfunction corresponding to λ1 and y2 an eigenfunction corresponding to λ2. Let α be the smallest zero of y1 and β the largest. If we can show that

then we can conclude that: Proposition 15.2.2. The number of zeros of y2 is greater than the number of zeros of y1. So suppose, by way of contradiction, that y2(x) > 0 for x > β where y1(β) = 0 and also y1(x) > 0 in this range. We have:

Our assumptions imply that

for N ≥ β +1 where

So we must have

Since y1 and y2 are decaying solutions,

for large enough N. So the previous inequality implies that which is impossible as both y1 and tend to zero as N → ∞. Also, if y1 has no zeros, then y2 must have at least one since y1 and y2 must be orthogonal. So we have proved the proposition. From the uniqueness of the decaying solution of −y"+vy = 0 proved above, we conclude that: and

Proposition 15.2.3. There is precisely one eigenfunction (up to a constant multiplicative factor) corresponding to any eigenvalue. Arrange the eigenfunctions according to increasing number of zeros. Let nk denote the number of zeros of the k-th eigenfunction. From the preceding we know that nk ≥ k.We will now prove that nk ≤ k, so the k-th eigenfunction has k zeros. 15.3 15.3.1

Using the Quadratic Form H (f , g) and Rayleigh-Ritz The Quadratic Form H (f , g)

Without loss of generality (by adding a constant if necessary), we may assume that H is positive, and so has an associated quadratic form Q whose domain D(Q) consists of those absolutely continuous functions f for which

and on this domain

Furthermore, if g ∈ D(H) then g ∈ D(Q) and for any f ∈ Q(H) we have

Rayleigh-Ritz. Let Ek be the space spanned by the first k eigenvectors y0, . . . , yk is its orthogonal −1 of H and its orthogonal complement, so that complement in D(Q). By Rayleigh-Ritz we know that λk is the minimum of Q(f,

f) as f ranges over all elements of E⊥∩ D(Q) with (f, f) = 1 and we know that f = yk is uniquely determined by this (up to multiple by z with |z| = 1). We use this to prove that yk has at most k zeros. Suppose not, so yk has r > k zeros. Let ϕ1 be the function which equals yk on the interval from −∞ to the leftmost zero of yk and equals zero elsewhere. Let ϕ2 = yk on the interval between this zero and the next and zero elsewhere, etc. This way we get r +1 functions, each equalling yk on an interval between zeros (or before the first and after the last). Using Rayleigh-Ritz to finish the proof: These functions are clearly independent so span a space F of dimension r +1 ≥ k +2. For each of these functions we have

But then F ∩ E ⊥k is at least two-dimensional, so the minimum given by Rayleigh-Ritz is achieved by at least two independent functions, which is a contradiction. 15.4

Perron-Frobenius-Krein-Rutman

The setup: positivity preserving and strictly positive operators. The KreinRutman theorem is a generalization to the Hilbert space of a piece of the PerronFrobenius theorem. The set up is the following: (X, ν) is a measure space, and A is a bounded self-adjoint operator on and Definition 15.4.1. A is positivity preserving if ϕ ≥ 0 a.e. ⇒ Aϕ ≥ 0 almost everywhere. A is positivity improving or is strictly positive if ϕ ≥ 0 a.e., ϕ ≠ 0, ⇒ Aϕ > 0 a.e. The Krein-Rutman theorem, an inequality. Let A be positivity preserving. Let u ∈ be a real valued function, so that Au is also real valued. Decompose u into its positive and non-positive parts, i.e.,

where u+ and u− are non-negative and

Then (Au+, u−) ≥ 0 being the scalar product of two non-negative functions and similarly (u+, Au−) ≥ 0. So

The Krein-Rutman theorem, 2. Now suppose that is an eigenvalue of A and hence the largest eigenvalue, and let u be a corresponding eigenvector. Then, by (∗),

So the middle inequalities must be equalities and hence

We get a contradiction unless either u+ or u− = 0. Replacing u by −u if necessary, we may assume that u ≥ 0. The Krein-Rutman theorem: Statement and proof of the theorem. Now assume that A is positivity improving. Let ϕ be any positive function. Then

implying that Au > 0 a.e. But u = λ−1 Au and hence u > 0 a.e. In particular, there cannot be another eigenvector with eigenvalue λ orthogonal to u. So we have proved: Theorem 15.4.1. Let A be a bounded positivity improving self-adjoint operator on with (maximal) eigenvalue Then λ is simple and the corresponding eigenvector can be chosen to be positive almost everywhere. Application to the ground state of a Hamiltonian H = H0 + v with discrete spectrum, 1. One way of having a positivity improving operator is if it is given by an integral kernel G where G(x, y) > 0 almost everywhere. In what follows, where I will be only sketchy, I will apply this remark to the resolvent of a Hamiltonian which is bounded from below, for example, where

For μ ≪ 0 the “Green’s function” G(x, y, μ) is the integral kernel of the operator (H0 + (v(x)−μ))−1, i.e., satisfies

Application to the ground state of a Hamiltonian, 2. The singularity of G at the diagonal, i.e., as is the same as that of the free Hamiltonian, i.e., of the solution of −Δε(x) = δ(x), which is of the form with cn > 0. (Let’s assume n ≥ 3.) Since (as solutions of an elliptic equation) G cannot achieve an interior maximum or minimum, we conclude that Proposition 15.4.1. G(x, y, μ) > 0 for x ≠ y. Application to the ground state of a Hamiltonian, 3. Now let α be the minimal eigenvalue of H and u a corresponding eigenvector, called a ground state. Consider, for μ 0 it is movinga way from the origin and u− represents an incoming wave. 16.3.3

The Jost Solutions

Now let V ≢ 0, the “perturbed problem” (PP). A natural first step when dealing with this PP is to see if the time dependency in the problem can again be separated out. If we assume that this separation is possible then we have

where

We would like to find solutions of these equations which behave like the solutions to the free equation at ±∞. More specifically, we would like to find solutions which have the following uniform asymptotic behavior: as x →+∞

and, as x →−∞

An application of variation of constants shows that these equations and conditions are equivalent to the integral equations

as can be verified directly. Solutions of these equations can (at least formally) be obtained by the Volterra-Born method of successively substituting the equation into itself. These Jost solutions have been extensively studied and their principal features are conveniently summarized in the following: Theorem 16.3.1. The Schrödinger equation

has Jost solutions as given above as satisfying the Volterra integral equations if

The Jost solutions ψ± and ϕ± are unique and can be obtained by the Born series. For each x ∈ ℝ the Jost solutions ψ+(x, k) and ϕ+(x, k) and their derivatives ψ+x (x, k) and are continuous with respect to k for (k ≥ 0 and analytic with respect to k for (k > 0. Similarly for ψ− and ϕ− for (k ≤ 0. Recall about second order ordinary differential equations. Asecond order ordinary differential equation for a function y can be made into a first order

system of differential equations by introducing (y1, y2) and setting y1 = y, y2 = y'. We now review an elementary fact about linear systems of first order differential equations: 16.3.4

A Brief Excursion about Wronskians

Jacobi’s formula for the differential of a determinant. Let F (A) = det A so F is a smooth function on the n2 dimensional space of n ×n square matrices. We want to compute dF. Cramer’s rule says that if we let B = (Bij) where Bij = (−1)i+j× (the determinant of the matrix with the i -th row and j -th column removed) then

In particular, looking at the i -th row, we obtain Laplace’s formula

for any i. By Leibnitz,

Now the elements of A involved in the computation of Bik do not include Aij so the second summand vanishes, and since the Aik are independent, the first sum is just Bij. So So introducing the standard names (so B T = Aadj) we obtain Jacobi’s formula

If A is invertible so that Aadj = (det A)A−1 this becomes

If A = eB we get

Wronskians. Let X be a solution of the matrix linear differential equation LX where L is a matrix valued function of t. Then by Jacobi

this last equation by properties of the trace. But XXadj = det X so we obtain

This can be solved as

so that det X is non-zero for all t if it is non-zero for any t. Notice that in the case of a first order system arising from a second order differential operator with no first order terms, the matrix L has the form which has trace identically 0. So in this case the Wronskian is independent of x. Getting back to the Schrödinger equation. The Wronskian of two solutions y1(x, k) and y2(x, k) of the Schrödinger equation is denoted by W(y1, y2) and defined (by our passage from a second order equation to a system of a first order equations) as

where the prime denotes differentiation with respect to x. Two solutions are linearly independent if and only if their Wronskian is non-zero, and it is sufficient to check this at a single x. Getting back to the Jost solutions of the Schrödinger equation. For the Jost solutions ψ± we have, as x →∞,

so ψ+ and ψ− are linearly independent, and as x →−∞

so ϕ+ and ϕ− are linearly independent. In particular, we have

In the case Q ≡ 0 we have ψ± = exp(±ikx), ϕ± = exp(∓ikx) so in this case the matrix of the c’s is The Jost solutions. In the general case, the left-hand side of (16.14) represents a solution of the Schrödinger equation which, by the properties of the left-hand side, reduces to exp(−ikx) as x →−∞, hence a wave moving from right to left. The right-hand side reduces to c11(k) exp(ikx)+c12(k) exp(−ikx) as x →+∞. So (16.14) is a solution of the Schrödinger equation which represents the scattering, by the potential Q(x), of a plane wave of amplitude c12(k) incident from x = +∞ and moving right to left. The scattering process gives rise to a reflected plane wave of amplitude c11 moving left to right toward x = +∞ and to a transmitted wave with unit amplitude moving right to left toward x = −∞. The usual normalization. It is customary to normalize this process so that the incident wave has unit amplitude in which case (16.14) is rewritten as

where

The subscript R refers to the fact that we are dealing with an incident wave from the right. The minus sign is included for later convenience. Similarly, (16.15) can be interpreted as a solution of the Schrödinger equation which represents the scattering, by a potential Q (x), of a plane wave incident from x = −∞. The process gives rise as before to a scattered and transmitted wave and normalizing as before we can write

where

The scattering matrix. Let

Then we can write the above equations as

where

The matrix S(k) is called the scattering matrix for the problem.

Chapter 17

Some Three-dimensional Computations 17.1

The Yukawa Potential in Three Dimensions

Consider the operator

given by

Once again, recall that the domain of H0 is taken to be those ϕ ∈ L2(ℝ3) for which the differential operator on the right, taken in the distributional sense, when applied to ϕ gives an element of L2(ℝ3). Also recall that the operator H0 has a fancy name. It is called the “free Hamiltonian of non-relativistic quantum mechanics.” Strictly speaking, we should include factors involving ħ and m. The Fourier transform is a unitary isomorphism of L2(ℝ3) into L2(ℝ3) and carries H0 into multiplication by ξ2 whose domain consists of those such that belongs to L2(ℝ3). The operators

form a one parameter group of unitary transformations whose infinitesimal generator in the sense of Stone’s theorem is the operator consisting of multiplication by ξ2 with domain as given above. [The minus sign before the i in the exponential is the convention used in quantum mechanics. So we write exp −it A for the one parameter group associated to the self-adjoint operator A. I apologize for this (rather irrelevant) notational change, but I want to make the notation in this section consistent with what you will see in physics books.] The operator of multiplication by ξ2 is self-adjoint, and this is how we proved that the operator H0 is a self-adjoint transformation. The operator of multiplication by ξ2 is clearly non-negative and so every point on the negative real axis belongs to its resolvent set. Let us write a point on the negative real axis as −μ2 where μ > 0. Then the resolvent of multiplication by ξ2 at such a point on the negative real axis is given by multiplication by − f where

We can summarize what we “know” so far as follows: 1. The operator H0 is self-adjoint. 2. The one parameter group of unitary transformations it generates via Stone’s theorem is

where Û(t) is multiplication by e−itξ 2. 3. Any point −μ2, μ > 0 lies in the resolvent set of H0 and

where m f denotes the operation of multiplication by f and f is as given above. 4. If g ∈ S and mg denotes multiplication by g, then the operator F −1mgF

consists of convolution by ğ. Neither the function e−itξ 2 nor the function f belongs to S, so the operators U (t) and R(−μ2, H0) can only be thought of as convolutions in the sense of generalized functions. 17.1.1

The Yukawa Potential and the Resolvent

We claim that in three dimensions the operator (λ2 I + H0)−1 is given by convolution with gλ where

Indeed, using spherical coordinates based in the direction of x, we have exp(−ix · y)dx

As θ doesn’t enter into the integrand, the integral with respect to θ just pulls out a factor of 2π. The integral with respect to ϕ is, up to a factor, the integral of a derivative yielding So

The above elementary computation is taken from Davies (p. 61) [4]. In most texts, this computation is done by residue calculus. The function Y μ or gμ is known as the Yukawapotential. Yukawa introduced this function in 1934 to explain the forces that hold the nucleus together. The exponential decay with distance contrasts with that of the ordinary electromagnetic or gravitational potential 1/r and, in Yukawa’s theory, accounts for the fact that the nuclear forces are short range. In fact, Yukawa introduced a “heavy boson” to account for the nuclear forces. The role of mesons in nuclear physics was predicted by brilliant theoretical speculation well before any experimental discovery.

The Fourier transform of 1/r in three dimensions. Up to overall constants, we have shown that the Fourier transform of (p2 + μ2)−1 is e−μx/r. Passing to the limit μ →0 we find that up to overall constants, the Fourier transform of 1/r is so, in momentum space (putting in the constants) we have Fock’s version of the hydrogen Hamiltonian:

in his famous 1935 paper on the SO(4) symmetry on the hydrogen Hamiltonian. For a translation of this article see Chapter 9 of the book [23] by Stephanie Singer. Extension of the domain of definition. Consider the operator G0(λ) = (H0 − λ2)−1 = −RH0(λ2). We know that it is defined for λ with positive imaginary part and is given by convolution with gλ, i.e.,

As an operator on L2(ℝ3) this is defined for . But if f has compact support, then the integral on the right makes sense for all values of λ and is holomorphic in its dependence on λ. So we have proved: Theorem 17.1.1. In ℝ3, the operator G0(λ) = −RH0(λ2), initially defined (as an operator on L2) for extends to an entire operator-valued function:

of λ in the whole complex plane. This theorem (and the proof we gave) is true on ℝn when n > 1 is odd where one also has an explicit formula for the resolvent. But as this formula in higher dimensions involves the Hankel functions, I won’t go into it. In dimension one we have a singularity at the origin due to the factor occurring in the expression for the resolvent. Some general comments: We know that for any self-adjoint operator A, its

resolvent R A(z) is defined on ℂ\R and satisfies the bound

Hence for we have [0,∞) we have

If the spectrum of A is contained in ℝ+ =

for

In particular, this is true for H0 in any dimension. Let us now consider the operator H0 + V where V is a bounded potential of compact support. If V is real (as we will usually assume), this is a self-adjoint operator with essential spectrum [0,∞) and a finite number of negative eigenvalues. So GV ≔ −RH0+V (λ2) is meromorphic in the upper half plane with poles on the imaginary axis at points where λ2 is a (negative) eigenvalue of H V ≔ H0 + V.We can get an expression for GV for large values of (λ valid even when V is complex valued. Indeed,

Now which is < 1 if is sufficiently large. So I +VG0(λ) is invertible (by the Neumann series) in this range. So multiplying the above equation by (I +VG0(λ))−1 gives

17.1.2

The Time Evolution of the Free Hamiltonian

The explicit calculation of the operator U (t) is slightly more tricky. The function is an “imaginary Gaussian,” so we expect its inverse Fourier transform to also be an imaginary Gaussian, and then we would have to make sense of convolution by a function which has absolute value one at all points. There are several ways to proceed. One involves integration by parts, and the method of stationary phase. Here I will follow Reed-Simon, vol. II, p. 59 [17] and add a little positive term to t and then pass to the limit. In other words, let α be a complex number

with positive real part and consider the function

This function belongs to S and its inverse Fourier transform is given by the function

(In fact, we verified this when α is real, but the integral defining the inverse Fourier transform converges in the entire half plane Re α > 0 uniformly in any Re and so is holomorphic in the right-half plane. So the formula for real positive α implies the formula for α in the half plane.) We thus have

Here the square root in the coefficient in front of the integral is obtained by continuation from the positive square root on the positive axis. For example, if we take α = +it so that −α = −i (t −i) we get

Here the limit is in the sense of L2. We thus could write

if we understand the right-hand side to mean the limit of the preceding expression. Actually, as Reed and Simon point out, if ϕ ∈ L1 the above integral exists for any t ≠ 0, so if ϕ ∈ L1 ∩ L2 we should expect that the above integral is indeed the expression for U (t)ϕ. Here is their argument: We know that

in the sense of L2 convergence as Here we use a theorem from measure theory which says that if you have an L2 convergent sequence you can choose a subsequence which also converges pointwise almost everywhere. So choose a subsequence of for which this happens. But then the dominated convergence theorem kicks in to guarantee that the integral of the limit is the limit of the integrals. To sum up: The function

is called the free propagator. For ϕ ∈ L1 ∩ L2

and the integral converges. For general elements ψ of L2 the operator is obtained by taking the L2 limit of the above expression for any sequence of elements of L1 ∩ L2 which approximate ψ in L2. Alternatively, we could interpret the above integral as the limit of the corresponding expression with t replaced by t −i∈.

Chapter 18

Bound States and Scattering States 18.1

Introduction

It is a truism in atomic physics or quantum chemistry courses that the eigenstates of the Schrödinger operator for atomic electrons are the bound states, the ones that remain bound to the nucleus, and that the “scattering states” which fly off in large positive or negative times correspond to the continuous spectrum. In terms of the Born interpretation of |ψ(x, t)|2 as the probability density of finding ψ is some region of space, if ψ is an eigenstate of H, so Hψ = Eψ, and Vt = exp(−i tH) is the time evolution corresponding to H, then |Vtψ|2 = |ψ|2 so the state is “bound.” The purpose of this chapter is to give a mathematical justification for the above truism about “scattering states.” The key result is due to Ruelle (1969), using ergodic theory methods. The ergodic theory used is limited to the mean ergodic theorem of von-Neumann that has a very slick proof due to F. Riesz (1939), which I shall give. In addition to the paper by Ruelle, the material in much of today’s lecture is taken from the two papers: W.O. Amrein andV.Georgescu, Helvetica Physica Acta 46 (1973) pp. 636–658, and W. Hunziker and I. Sigal, J. Math. Phys. 41 (2000) pp. 3448–3510. There is also an important paper by Enss, Comm. Math. Phys. 61 (1978) pp. 285–291. So in many texts, the main theorem relating bound and scattering states to the point and the continuous spectrum, respectively, is known as the RAGE theorem. Here are the tools we will use: • The mean ergodic theorem, stated and proved below.

• •

•

18.2

The Kato-Rellich theorem, recalled below. If H is a self-adjoint operator on a Hilbert space with no point spectrum, then 0 is not an eigenvalue of H ⊗ I − I ⊗ H on proved below from the projection measure version of the spectral theorem. A compact operator on a separable Hilbert space is the norm limit of finite rank operators. We proved this in Chapter 10. The Mean Ergodic Theorem

We will need the continuous time version: Let H be a self-adjoint operator on a Hilbert space and let

be the one parameter group it generates (by Stone’s theorem). von Neumann’s mean ergodic theorem asserts that Theorem 18.2.1. For any

the limit

exists, call it g, and Hg = 0. Alternative form of the theorem. Clearly, if H f = 0, then Vt f = f for all t and the above limit exists trivially and is equal to f. If f is orthogonal to the image of H, i.e., if

then f ∈ Dom(H∗) = Dom(H) and H∗ f = H f = 0. So if we decompose into the zero eigenspace of H and its orthogonal complement, we are reduced to the following version of the theorem which is the one we will actually use: Theorem 18.2.2. Let H be a self-adjoint operator on a Hilbert space and assume that H has no eigenvectors with eigenvalue 0, so that the image of H is dense in Let Vt= exp(−i tH) be the one parameter group generated by H. Then

for all First, some details about Stone’s theorem. By the properties of the functional calculus we know that Vs+t = Vs ◦ Vt and so that is a one parameter group of unitary transformations. Proposition 18.2.1. Vt is strongly continuous. This means that for any

we have

We know this from our discussion of semi-groups, but I will give a proof from the spectral theorem: Proof.

By the dominated convergence theorem this approaches 0 as t → t0. The derivative of V t at 0. Proposition 18.2.2. The limit

exists if and only if ψ ∈ D(H) in which case

We also know this from our discussion of infinitesimal generators of semigroups, but again, I give a proof from the spectral theorem.

Proof. If ψ ∈ D(H), then

again by the dominated convergence theorem, since |eitλ −1| ≤ |tλ|. If we define the operator B by setting Bψ equal to the limit and with domain D(B) equal to the set of ψ where this limit exists, then for ϕ,ψ ∈ D(B) we have

So if we set A = i B then A is symmetric. Also, the domain of A equals the domain of B, and by the first part of the argument D(H) ⊂ D(B). But a selfadjoint operator cannot have any strict symmetric extensions, so A = H. Indeed, if H ⊂ A in the sense that D(H) ⊂ D(A) and Hϕ = Aϕ for ϕ ∈ D(H) then from the definition of the adjoint, A∗ ⊂ H ∗ (this for any pair of densely defined operators). If H is self-adjoint and A is symmetric (so A ⊂ A∗) then

This also shows that Vt (D(H)) ⊂ D(H) and for ϕ ∈ D(H) we have

Theorem 18.2.3 (Mean ergodic theorem). Let H be a self-adjoint operator on a Hilbert space and assume that H has no eigenvectors with eigenvalue zero, so that the image of H is dense in Let Vt= exp(−itH) be the one parameter group generated by H. Then

for all Proof. If h = −i Hg then

so

By hypothesis, for any such that

we can, for any > 0, find an h of the above form

By then choosing T sufficiently large we can make the second term less than 18.3

Recall: The Kato-Rellich Theorem

Cores, relative bounds. A symmetric operator T is called essentially selfadjoint if its closure is self-adjoint. If A is a self-adjoint operator with domain D(A), a subspace D ⊂ D(A) is called a core for A if the closure of the restriction of A to D is A. A-bounded operators. Let A and B be densely defined operators on a Hilbert space H. We say that B is A-bounded if • D(B) ⊃ D(A), and • There exist real numbers a and b such that

Recall that if

then (9.1) holds. On the other hand, if (9.1) holds, then

Writing ab = (a)(b−1) we get

So (9.1) implies (9.2) with a replaced by a +and b replaced by b+−1. Thus the infimum of a over all (a, b) such that (9.1) holds is the same as the infimum of a over all (a, b) such that (9.2) holds. This common infimum is called the relative bound of B with respect to A. If this relative bound is 0 we said that B is infinitesimally small with respect to A. In verifying (9.1) or (9.2) it is sufficient to do so for all ϕ belonging to a core of A. The following theorem was proved by Rellich in 1939 and was extensively used by Kato in the 1960’s and is known as the Kato-Rellich theorem. Theorem 18.3.1 (Kato-Rellich theorem). Let A be a self-adjoint operator and B a symmetric operator which is relatively A-bounded with relative bound a < 1. Then A + B is self-adjoint on D(A) and is essentially self-adjoint on any core of A. If A is bounded below by M then A + B is bounded below by

for any (a, b) for which (9.1) holds. We proved this in Chapter 9. 18.4

Characterizing Operators with Purely Continuous Spectrum

Suppose that A is a self-adjoint operator with only continuous spectrum. Let

be its spectral resolution. For any ψ ∈ the function

is continuous. It is also a monotone increasing function of μ. For any> 0 we can find a sufficiently negative a such that |(Eaψ,ψ)| 0 such that

for all μ ∈ ℝ. So

This says that the measure of the band of width δ about the diagonal has measure less that ∈ Letting δ shrink to 0 shows that the diagonal line has measure zero. We can restate this lemma more abstractly as follows: Consider the Hilbert space (the completion of the tensor product The Eλ and Eμ determine a projection valued measure Q on the plane with values in The spectral measure associated with the operator A⊗ I − I ⊗ A is then Fρ ≔ Q({(λ, μ)| λ−μ < ρ}). So an abstract way of formulating the lemma is Proposition 18.4.1. A has continuous spectrum if and only if 0 is not an eigenvalue (If ψ is an eigenvector of A then ψ ⊗ψ is an eigenvector of A⊗ I − I ⊗ A with eigenvalue zero, giving the trivial converse of what we proved above.) 18.5

The RAGE Theorem

A decomposition of the spectrum. Let H be a self-adjoint operator on a separable Hilbert space and let Vt be the one parameter group generated by H so

Let

be the decomposition of into the subspaces corresponding to pure point spectrum and continuous spectrum of H. Increasing projections. Let {Fr }, r = 1, 2, . . . be a sequence of self-adjoint projections with

(In the application we have in mind we will let and take Fr to be the projection onto the completion of the space of continuous functions supported in the ball of radius r centered at the origin, but for now our considerations will be quite general.) We let be the projection onto the subspace orthogonal to the image of Fr so

18.5.1

The Spaces M0 and M∞

Let

and

In terms of our application, we want to think of M0 as consisting of states which “stay in a bounded neighborhood of the origin for all time,” and of M∞ as consisting of states which “fly off to infinity.” Proposition 18.5.1. The following hold:

1.

M0 and M∞ are linear subspaces of

2.

The subspaces M0 and M∞ are closed.

3.

M0 is orthogonal toM∞.

The following inequality will be used repeatedly: For any

where the last equality is the theorem of Appolonius. M0 is a linear subspace of Proof. Let f1, f2 ∈M0. Then for any scalars a and b and any fixed r and t we have

by (18.5). Taking separate sups over t on the right side and then over t on the left shows that

for fixed r. Letting r → ∞ then shows that a f1 + bf2 ∈M0. M∞ is a linear subspace of H.

Proof. Let f1, f2 ∈M∞. For fixed r we use (18.5) to conclude that

Each term on the right converges to 0 as T →∞proving that a f1 +bf2 ∈M∞. M0 is closed. Proof.

∈ and suppose that for all n > N. This implies that

Given

choose N so that

for all t and r since Vt is unitary and I − Fr is a contraction. Then

for all n > N and any fixed r. We may choose r sufficiently large so that the second term on the right is also less that This proves that f ∈M0. M∞ is closed. Proof. Let fn ∈M∞ and suppose that fn → f. Given > 0 choose N so that for all n > N. Then

Fix n. For any given r we can choose T0 large enough so that the second term on the right is This shows that for any fixed r we can find a T0 so that

for all T > T0, proving that f ∈M ∞. M0 is orthogonal to M∞. Proof. Let f ∈ M0and g ∈M∞ both ≠ 0. Then

where we used the Cauchy-Schwarz inequality in the last step. For any> 0 we may choose r so that

for all t. We can choose a T such that

Plugging back into the last inequality shows that

Since this is true for any > 0 we conclude that

Proof. Suppose Hf = E f. Then

Recall that

But we are assuming that tends to 0 proving that f ∈M0.

Proof. We know that

in the strong topology. So this last expression

Since

we have

Summary of what we have proved so far: {Fr }, r = 1, 2, . . . is a sequence of self-adjoint projections with

and

Then: 1. M0 and M∞ are linear subspaces of 2. The subspaces M0 and M∞ are closed. 3. M0 is orthogonal to M∞.

The above facts are valid without any assumptions whatsoever relating H to the Fr. The only place where we used H was in the proof of 4 where we used the fact that if f is an eigenvector of H then it is also an eigenvector of Vt and so we could pull out a scalar. The goal is to impose sufficient relations between H and the Fr so that

If we prove this then

implies that

and then the fact that M0 is orthogonal to M∞ says that

Then

18.5.2

gives

Using the Mean Ergodic Theorem

Recall that the mean ergodic theorem says that if Ut is a unitary one parameter group acting without (non-zero) fixed vectors on a Hilbert space then

for all ψ ∈ Let

We know, from our discussion above, that H ⊗ I − I ⊗ H does not have zero as an eigenvalue acting on may apply the mean ergodic theorem to conclude that

for any We have

We conclude that

Indeed, if identically zero. 18.5.3

this follows from the above, while if

the integrand is

The Amrein-Georgescu Theorem

We continue with the previous notation, and let denote orthogonal projection. We let Sn and S be a collection of bounded operators on such that • • • •

[Sn, H] = 0, Sn → S in the strong topology, The range of S is dense in and Fr Sn Ec is compact for all r and n.

Theorem 18.5.1 (Amrein-Georgescu). Under the above hypotheses

Proof. Since M∞ is a closed subspace of , to prove that (18.6) holds, it is enough toprove that

for some set D which is dense in Since S leaves the spaces and invariant, the fact that the range of S is dense in by hypothesis says that is dense in So we have to show that

for any fixed r. We may assume f = 0. To prove that for any fixed r :

Let > 0 be fixed. Choose n so large that

Any compact operator in a separable Hilbert space is the norm limit of finite rank operators. So we can find a finite rank operator Q N such that

Writing g = (S − Sn) f + Sn f we conclude that

To say that QN is of finite rank means that there are gi, hi ∈ H, 0, . . . , N T0.

Kato Potentials

Definition of Kato potentials. Let X = Rn for some n. A locally L2 real valued function V on X is called a Kato potential if for any α > 0 there is a β = β(α) such that

for all Clearly the set of all Kato potentials on X form a real vector space. Examples of Kato potentials. V ∈ L2(ℝ3). Suppose that X = ℝ3 and V ∈ L2(X). We claim that V is a Kato potential. Indeed,

So we will be done if we show that for any a > 0 there is a b > 0 such that

By the Fourier inversion formula we have

where denotes the Fourier transform of ψ. Now the Fourier transform of H0ψ is the function

where

denotes the Euclidean norm of ξ.

To prove:

Now

Since belongs to the Schwartz space S, the function

belongs to L2 as does the function in three dimensions. Let λ denote the function By the Cauchy-Schwarz inequality we have

where

For any r > 0 and any function ϕ ∈ S let ϕr be defined by

Then

Applied to ψ this gives

By Plancherel

This shows that any V ∈ L2(ℝ3) is a Kato potential. V ∈ L∞(X). Indeed

If we put these two examples together we see that if V = V1 + V2 where V1 ∈ L2(ℝ3) and V2 ∈ L∞(ℝ3) then V is a Kato potential. The Coulomb potential. The function

on R3 can be written as a sum V = V1 + V2 where V1 ∈ L2(ℝ3) and V2 ∈ L∞(ℝ3) and so is Kato potential. Kato potentials from subspaces. Suppose that X = X1 ⊕ X2 and V depends only on the X1 component where it is a Kato potential. Then Fubini’s theorem implies that V is a Kato potential if and only if V is a Kato potential on X 1. So if X = ℝ3N and we write x ∈ X as x = (x1, . . . , xN) where xi ∈ ℝ3 then

are Kato potentials as are any linear combination of them. So the total Coulomb potential of any system of charged particles is a Kato potential. By the above, the restriction of this potential to the subspace

is a Kato potential. This is the “atomic potential” about the center of mass. Applying the Kato-Rellich method to Kato potentials. Theorem 18.6.1. Let V be a Kato potential. Then

is self-adjoint with domain D = Dom(H0) and is bounded from below. Furthermore, we have an operator bound

where

As a multiplication operator, V is closed on its domain of definition consisting of all ψ ∈ L2 such that Vψ ∈ L2. Since is a core for H0, we can apply the Kato condition (18.8) to all ψ ∈ Dom(H0). Thus H is defined as a symmetric operator on Dom(H0). For Re z < 0 the operator (z I − H0)−1 is bounded. So for Re z < 0 we can write

By the Kato condition (18.8) we have

If we choose α < 1 and then Re z sufficiently negative, we can make the righthand side of this inequality < 1. For this range of z we see that R(z, H) = (z I − H)−1 is bounded so the range of z I − H is all of L2. This proves that H is self-adjoint and that its resolvent set contains a half plane Re z ≪ 0 and so is bounded from below. Also, for ψ ∈ Dom(H0) we have

so

which proves (18.9). Using the inequality H 0 ≤ aH + b (18.9). Proposition 18.6.1. Let H be a self-adjoint operator on L2(X) satisfying (18.9) for some constants a and b. Let f ∈ L∞(X) be such that f (x)→0 as x →∞. Then for any z in the resolvent set of H the operator

is compact, where, as usual, f denotes the operator of multiplication by f.

Proof. Let as usual, and let g ∈ L∞(X∗) so the operator g(p) is defined as the operator which send ψ into the function whose Fourier transform is . The operator f (x)g(p) is the norm limit of the operators fngn where fn is obtained from f by setting fn = 1Bn f where Bn is the ball of radius n about the origin, and similarly for g. The operator fn(x)gn(p) is given by the square integrable kernel

and so is compact. Hence f (x)g(p) is compact. We will take

The operator (1+ H0)R(z, H) is bounded. Indeed, by (18.9)

So

is compact, being the product of a compact operator and a bounded operator. 18.7

Ruelle’s Theorem

Let us take H = H0 + V where V is a Kato potential. Let Fr be the operator of multiplication by 1Br so Fr is projection onto the space of functions supported in the ball Br of radius r centered at the origin. Take S = R(z, H), where z has sufficiently negative real part. Then FrSEc is compact, being the product of the operator Fr ℝ (z, H) (which is compact by the preceding proposition) and the bounded operator Ec. Also the image of S is all of D(H) which is dense in . So we may apply the Amrein Georgescu theorem to conclude that M0 = Hp and M∞ = c.

Chapter 19

The Exponential Decay of Eigenstates 19.1

Introduction

We proved Ruelle’s theorem which asserts that for the Schrödinger operator

where V is a Kato potential, the point spectrum corresponds to the bound states and the continuous spectrum to the scattering states. But in cases where we have explicit solutions to the Schrödinger equation, there is much more precise information: the bound states decay exponentially at infinity. We saw this explicitly for the case of the square well in Chapter 9. Indeed, for the case of the square well we were able to compute the eigenvalues and eigenfunctions explicitly because the elements of the domain of H were (almost everywhere) continuous functions with continuous derivatives, a fact that is not true in higher dimensions. Recall that the eigenfunctions for the square well looked like: The purpose of this chapter is to present a very watered down version of a general theory of such exponential decay which was developed by my old friend Shmuel Agmon, cf. his book [1]. All the material in this chapter is taken from [8]. Quantum tunneling. We are studying the Schrödinger equation for H = H0 + V, and suppose that lim inf and that V is sufficiently deep to

support a bound state (i.e., an eigenfunction) at energy E, so

Furthermore, we suppose that outside some compact set W(E), we have E − V < 0. A classical particle with energy E is forbidden to enter Rn \ W(E) by the conservation of energy. The quantum mechanical wave function, ψ, however, penetrates this region. Recall that this effect is called quantum tunneling. We want to show that ψ dies exponentially at ∞in a sense that we will make precise. Here is a diagram of “quantum tunneling”:

A potential well with the classically forbidden outside the vertical dashed lines

19.2

The Agmon Metric

Definition 19.2.1 (The Agmon “scalar product”). Let ξ, η ∈ Tx (ℝn). Define

where (ξ, η) is the usual Euclidean scalar product and f (x)+ = max(0, f (x)). This is a degenerate scalar product, but we can use it to define the “length” of a curve as

where 19.2.1

is the usual Euclidean length. The Agmon Distance

Define the Agmon distance ρE (x, y) between x, y ∈ ℝn to be the infimum of L(γ) over all piecewise differentiable curves γ joining x to y. It is clear that the triangle inequality holds. Proposition 19.2.1. The distance function ρ is locally Lipschitz continuous and hence is differentiable almost everywhere in x and in y. Proof. Fix x and let y, z ∈ ℝn and

so

which we can write as

So if we let BR(y) denote the ball of radius R centered at y and CR(y) ≔ maxw∈BR(y) Cy,w we have

for w, z ∈ B R(y). This shows that ρE is Lipschitz and hence differentiable almost everywhere in its second variable and by symmetry, also in its first. From the inequality

it also follows (letting R → 0) that at points where ρE is differentiable in its second variable we have

19.3

Agmon’s Theorem

Let H = H0 + V be a self-adjoint operator and H ψ = Eψ where

is a compact subset of ℝn. Let

Theorem 19.3.1. For any > 0 ∃ c, a constant > 0, such that

19.3.1 Conjugation and Commutation of H 0 by a Multiplication Operator For the proof, we need some conjugation and commutation results: We begin with the fact that

In this equation f is a smooth function considered as the operator of multiplication by f and∇ f is the operator of multiplication by the vector valued function∇ f, and [A, B] denotes the commutator, [A, B] ≔ AB − BA of operators. Indeed, since ∇ is a first order differential operator, Leibnitz tells us that ∇(fg) = (∇ f)g + f∇g so

as desired.

So

so

An interlude. In a general associative ring we let [A, B] ≔ AB − BA for any two elements A, B in the ring, and we let ad A denote the linear transformation If u is an element of the ring with a two-sided inverse, we denote by Ad u the automorphism of the ring defined by

In a general ring we may not be able to make sense of the exponential function. That is, there might be no sense in which the infinite series

converges, and if the ring does not contain the rational numbers, the coefficients will not make any sense. We might also have trouble making sense of the operator series

But a fundamental formula that we expect to hold when both exponentials do make sense is

For example, in our situation, if A denotes multiplication by the function f, then

exp A would be multiplication by the function ef. On the other hand, if B is a linear differential operator of order k, then it is easy to check that (ad f)(B) is a linear differential operator of order (at most) k −1 and hence (ad f)m B = 0 for m > k. So the series (exp ad f)B is actually a finite sum. In fact, therefore, (19.2) follows from (19.1). 19.4

Some Inequalities

Since multiplication operators commute, the same formula (19.2) applies with H0 replaced by H. So if we let Hf ≔ e f He−f we have

For a smooth real valued function f of compact support and any smooth ϕ of compact support we have

We have (ϕ, H0ϕ) ≥ 0 since H0 is a non-negative operator. We can use integration by parts to write the second term as

which is the complex conjugate of the negative of the third term. So we have

We initially proved this for f and ϕ smooth and of compact support. But the quadratic forms on both sides of this inequality make sense for ϕ of compact support and f Lipschitz. By approximation, this inequality holds for f Lipschitz. Suppose that ∇ f satisfies

and that ϕ satisfies

where

Here 0 < < 1 and δ > 0. Then the right-hand side of (19.4) is ≥ δ║ ϕ║2. So under the above hypotheses about f and ϕ we have

We are now going to get an alternative expression for the left-hand side of (19.7) under the assumptions that • • • •

f is real and bounded, ϕ = η · ef ·ψ where H ψ = E ψ and ∇η is a smooth real valued function with compact support.

Under these assumptions, using

and (H − E)ψ = 0, we get

So we have proved that

Since ∇η is of compact support, we can integrate the second term on the right in (19.8) by parts to obtain

We can bring the last term on the right in (∗) over to the left and use the fact that η and f are real to conclude that

If we now add the first term on the right-hand side of (19.8) we obtain

We are now going to choose ηand f so that (19.9) together with the inequality

prove Agmon’s theorem. 19.5

Completion of the Proof

A choice of η and f. Let

which is the same notation as above. Let

Let η be a smooth function which is identically 0 on AE,δ and is identically 1 on FE,2δ. Our assumption about V says that ∇η has compact support. Set

where ρE (x) = ρ(x, 0) is the Agmon distance of x to the origin.

Classically forbidden region F E2δ and allowed region AE, δ

Let fα for α ≥ 0 be defined by

Notice that

that fα is bounded for α > 0 and that limα→0 fα = f0 = g. Our task is to prove that

when ψ is an eigenvector of H with eigenvalue E. We can write

The set is compact, and g is continuous, so e2g is bounded on this set, and hence the second integral is bounded by some constant times So it is enough to prove that

is bounded by some constant times ║ψ║2. By the monotone convergence theorem, it is enough to show that there is a constant c independent of α such that

We can apply

and then

to ϕ = η · efα ·ψ and conclude that

where

Pointwise exponential bounds. Under smoothness hypotheses about V, Agmon’s theorem (which gives a sort of “averaged” exponential bound) can be converted to a pointwise exponential bound. This is so because the regularity of V will imply the regularity of any eigenfunction of H. This is an example of what is known as elliptic engineering via a method due to Hermann Weyl. Roughly speaking, it says that if V has k continuous derivatives then any eigenfunction has k +2 continuous derivatives. From this one can conclude that for sufficiently smooth V (and the hypotheses of Agmon’s theorem that we have proved) that we get an exponential bound of the form

For the proof see [8] pp. 34–36.

Chapter 20

Lorch’s Proof of the Spectral Theorem After a hiatus from proofs of the spectral theorem, I will present another one— Lorch’s proof as it appears in his book “Spectral Theory.” A key tool is the Riesz-Dunford calculus. One pleasant by-product is a proof of Gelfand’s formula for the spectral radius which does not depend on the principle of uniform boundedness. But our main interest in Lorch’s proof is that it gives good motivation via elementary complex analysis to Stone’s formula for the spectral measure. 20.1

The Riesz-Dunford Calculus

We will develop this “calculus” over the next few paragraphs. Suppose that we have a continuous map defined on some open set of complex numbers, where Sz is a bounded operator on some fixed Banach space; by continuity, we mean continuity relative to the uniform metric on operators. If C is a continuous piecewise differentiable (or more generally any continuous rectifiable) curve lying in this open set, and if is a piecewise smooth (or continuous, rectifiable) parametrization of this curve, then the map is continuous. 20.1.1

The Riemann Integral over a Curve

For any partition 0 = t0 ≤ t1 ≤· · · ≤ tn = 1 of the unit interval we can form the

Cauchy approximating sum

The usual proof of the existence of the Riemann integral shows that this tends to a limit as the mesh becomes more refined and the mesh distance tends to 0. We denote the limit by

This notation is justified because the change of variables formula for an ordinary integral shows that this value does not depend on the parametrization, but only on the orientation, of C. Application to the resolvent. We are going to apply this to Sz = Rz, the resolvent of an operator, and the main equations we shall use are old friends, the resolvent equation:

and the power series for the resolvent:

We proved that the resolvent of a self-adjoint operator exists for all non-real values of z. But a lot of the theory goes over for the resolvent

where T is an arbitrary operator on a Banach space, so long as we restrict ourselves to the resolvent set, i.e., the set where the resolvent exists as a bounded operator. So, following [14], we first develop some facts about integrating the resolvent in the more general Banach space setting (where our principal application will be to the case where T is a bounded operator). For example, suppose that C is a simple closed curve contained in the disk of convergence about z of the above power series for Rw. Then we can integrate the

series term by term. But

for all n ≠ −1 so

By the usual method of breaking up any deformation into a succession of small deformations and then breaking up any small deformation into a sequence of small “rectangles” we conclude: Theorem 20.1.1. If two curves C0 and C1 lie in the resolvent set and are homotopic by a family Ct of curves lying entirely in the resolvent set, then

Here are some immediate consequences of this elementary result: Liouville’s argument. Suppose that T is a bounded operator and

Then

exists because the series in parentheses converges in the uniform metric. In other words, all points in the complex plane outside the disk of radius lie in the resolvent set of T. From this it follows that the spectrum of any bounded operator cannot be empty (if the Banach space is not {0}). (Recall the spectrum is the complement of the resolvent set.) Indeed: Proof. Indeed, if the resolvent set were the whole plane, then the circle of radius zero about the origin would be homotopic to a circle of radius via a homotopy lying entirely in the resolvent set. Integrating Rz around the circle of radius zero gives 0. We can integrate around a large circle using the above power series. In performing this integration, all terms vanish except the first which give 2πi I by the usual Cauchy integral (or by direct computation). Thus 2π I = 0 which is impossible in a non-zero vector space.

Here is another very important (and easy) consequence of the preceding theorem: Theorem 20.1.2. Let C be a simple closed rectifiable curve lying entirely in the resolvent set of T. Then

is a projection which commutes with T, i.e.,

Proof. Choose a simple closed curve C' disjoint from C but sufficiently close to C so as to be homotopic to C via a homotopy lying in the resolvent set. Thus

and so

where we have used the resolvent equation

We write this last expression as a sum of two terms,

Choose C' to lie entirely inside C. Then the first expression above is just (2πi) C' Rwdw while the second expression vanishes, all by the elementary Cauchy integral of 1/(z −w). Thus we get

or P2 = P. So P is a projection. It commutes with T because it is an integral whose integrand Rz commutes with T for all z. The same argument proves: Theorem 20.1.3. Let C and C be simple closed curves each lying in the resolvent set, and let P and P' be the corresponding projections given by (20.1). Then PP= 0 if the curves lie exterior to one another while P P'= P'if C' is interior to C. A decomposition. Let us decompose our Banach space B as B = B'⊕ B" where

are the images of the projections P and I − P where P is given by

Each of these spaces is invariant under T and hence under Rz because PT = TP and hence PRz = Rz P. For any transformation S commuting with P let us write

so that S' and S"are the restrictions of S to Band B", respectively. For example, we may consider For x'∈ B' we have . In other words, is the resolvent of T' (on B') and similarly for R"z. So if z is in the resolvent set for T, it is in the resolvent set for T' and T". Conversely, suppose that z is in the resolvent set for both T' and T". Then

there exists an inverse A1 for zI' − T' on Band an inverse A2 for zI" − T" on Band so A1 ⊕ A2 is the inverse of zI − T on B = B'⊕ B. So a point belongs to the resolvent set of T if and only if it belongs to the resolvent set of T' and T". Since the spectrum is the complement of the resolvent set, we can say that a point belongs to the spectrum of T if and only if it belongs either to the spectrum of T or T :

We now show that this decomposition is in fact the decomposition of Spec(T) into those points which lie inside C and outside C. So we must show that if z lies exterior to C then it lies in the resolvent set of T'. This will certainly be true if we can find a transformation A on B which commutes with T and such that

for then A' will be the resolvent at z of T' . Now

so

We have thus proved: Theorem 20.1.4. Let T be a bounded linear transformation on a Banach space and C a simple closed curve lying in its resolvent set. Let P be the projection given by

and

the corresponding decomposition of B and T. Then Spec (T') consists of those

points of Spec(T) which lie inside C and Spec(T") consists of those points of Spec(T) which lie exterior to C. In particular, if P is the projection associated to a simple closed curve lying in the resolvent set, as above, then: Proposition 20.1.1. P = 0 if and only if the interior of C belongs to the resolvent set and P = I if and only if the spectrum of T lies entirely in the interior of C. 20.1.2

Lorch’s Proof of Gelfand’s Formula for the Spectral Radius

As another special case of the theorem, consider the case where the curve C is the unit circle. So we are assuming that the unit circle lies entirely in the resolvent set of T and we want to compute the integral Lorch does this by choosing the Riemman approximation corresponding to the division of the circle into n parts by the points α0, α1, α2, . . . ,αn−1 where α = exp(2πi)/n). So we want to consider the sum

Using the partial fraction expansion for

the above expression simplifies to

Now α = exp(2πi/n) so n(α −1)→2πi as n →∞so we see that the projection P associated to the unit circle (assuming that it lies in the resolvent set) is

Proof. In particular, suppose that the spectral radius of T is < 1 so the entire spectrum is in the interior of the unit circle, so limn→∞(I − Tn)−1 = I and hence T n →0. Hence so for large n we have so lim Now let B be any bounded operator with spectral radius rB and let T ≔ (rB +) −1B. We apply the preceding to T. Letting →0 we conclude that lim and hence we obtain Gelfand’s formula

20.1.3

Stone’s Formula

We now begin to have a better understanding of Stone’s formula: Suppose A is a self-adjoint operator. We know that its spectrum lies on the real axis. If we draw a rectangle whose upper and lower sides are parallel to the axis, and whose vertical sides do not intersect Spec(A), we would get a projection onto a subspace M of our Hilbert space which is invariant under A, and such that the spectrum of A when restricted to M lies in the interval cut out on the real axis by our rectangle. The problem is how to make sense of this procedure when the vertical edges of the rectangle might cut through the spectrum, in which case the integral

might not be defined. This is resolved by the method of Lorch (the exposition is taken from his book), which we explain in the next section. 20.2 20.2.1

Lorch’s Proof of the Spectral Theorem The Point Spectrum

We now let A denote an arbitrary (not necessarily bounded) self-adjoint transformation. Recall that we say that λ belongs to the point spectrum of A if there exists an x ∈ D(A) such that x ≠ 0 and Ax = λx; in other words, if λ is an eigenvalue of A. Notice that eigenvectors corresponding to distinct eigenvalues are orthogonal: if Ax = λx and Ay = μy, then

implying that (x, y) = 0 if λ ≠ μ. Also, the fact that a self-adjoint operator is closed implies that the space of eigenvectors corresponding to a fixed eigenvalue is a closed subspace of . We let Nλ denote the space of eigenvectors corresponding to an eigenvalue λ. 20.2.2

Operators with Pure Point Spectrum

We say that A has pure point spectrum if its eigenvectors span , in other

words, if

where the λi range over the set of eigenvalues of A. Suppose that this is the case. Then let

where this denotes the Hilbert space direct sum, i.e., the closure of the algebraic direct sum. Let Eλ denote projection onto Mλ.

where this denotes the Hilbert space direct sum, i.e., the closure of the algebraic direct sum. Let Eλ denote projection onto Mλ. Then it is immediate that the Eλ satisfy the conditions of the spectral theorem. We thus have a proof of the spectral theorem for operators with pure point spectrum. 20.2.3

Partition into Pure Types

Now consider a general self-adjoint operator A, and let

(Hilbert space direct sum) and set

The space and hence the space 2 are invariant under A in the sense that A maps D(A) ∩ 1 to 1 and similarly for 2. We let P denote orthogonal projection onto 1 so I − P is orthogonal projection onto 2. We claim that

Suppose that x ∈ D(A). We must show that Px ∈ D(A) for x = Px +(I − P)x is a decomposition of every element of D(A) into a sum of elements of D(A) ∩ 1 and D(A) ∩ 2. By definition, we can find an orthonormal basis of 1 consisting of eigenvectors ui of A, and then

The sum on the right is (in general) infinite. Let y denote any finite partial sum. Since eigenvectors belong to D(A) we know that y ∈ D(A). We have

since x − y is orthogonal to all the eigenvectors occurring in the expression for y. We thus have

From this we see (aswe let the number of terms in y increase) that both y converges to Px and the Ay converge. Hence Px ∈ D(A), proving our contention. Let A1 denote the operator A restricted to P[D(A)] = D(A) ∩ 1 with similar notation for A2. We claim that A1 is self-adjoint (as is A2). Clearly, D(A1) ≔ P(D(A)) is dense in 1, for if there were a vector y ∈ 1 orthogonal to D(A1) it would be orthogonal to D(A) in which is impossible. Similarly, D(A2) ≔ D(A) ∩ 2 is dense in 2. Now suppose that y1 and z1 are elements of 1 such that

Since A1x1 = Ax1 and x1 = x − x2 for some x ∈ D(A), and since y1 and z1 are orthogonal to x2, we can write the above equation as

which implies that y1 ∈ D(A) ∩ 1 = D(A1) and A1y1 = Ay1 = z1. In other words, A1 is self-adjoint. Similarly, so is A2. We have thus proved:

Theorem 20.2.1. Let A be a self-adjoint transformation on a Hilbert space . Then

with self-adjoint transformations A1 on 1 having pure point spectrum and A2 on 2 having no point spectrum such that

and

We have proved the spectral theorem for a self-adjoint operator with pure point spectrum. Our proof of the full spectral theorem will be complete once we prove it for operators with no point spectrum. 20.2.4

Completion of Lorch’s Proof

In this subsection we will assume that A is a self-adjoint operator with no point spectrum, i.e., no eigenvalues. Let λ < μ be real numbers and let C be a closed piecewise smooth curve in the complex plane which is symmetrical about the real axis and cuts the real axis at a non-zero angle at the two points λ and μ (only). Let m > 0 and n > 0 be positive integers, and let

In fact, we would like to be able to consider the above integral when m = n = 0, in which case it should give us a projection onto a subspace where λI ≤ A ≤ μI . Unfortunately, if λ or μ belong to Spec(A) the above integral need not converge with m = n = 0. However, we do know that so that the blowup in the integrand at m n λ and μ is killed by (z −λ) and (μ−z) when m ≥ 1 and n ≥ 1 since the curve makes a non-zero angle with the real axis. Since the curve is symmetric about the real axis, the (bounded) operator K λμ(m, n) is self-adjoint. Furthermore, modifying the curve C to a curve Clying inside C, again intersecting the real axis only at the points λ and μ and having these intersections at non-zero angles does

not change the value: Kλμ(m, n).

We will now prove a succession of facts about K λμ(m, n):

Proof. Calculate the product using a curve C' for K λμ(m', n') as indicated above. Then use the functional equation for the resolvent and Cauchy’s integral formula exactly as in the proof of the theorem about projections that we proved before:

which we write as a sum of two integrals, the first giving (2πi)2Kλμ(m + m', n + n') and the second giving zero. A similar argument (similar to the proof of the theorem about curves exterior to each other) shows that

Proposition 20.2.1. There exists a bounded self-adjoint operator L λμ(m, n) such that

Proof. The function is defined and holomorphic on the complex plane with the closed intervals (−∞, λ] and [μ,∞) removed. The integral

is well defined since, if m = 1 or n = 1, the singularity is of the form at worst, which is integrable. Then the proof of (20.5) applies to prove the proposition. For each non-real z we know that Rz x ∈ D(A). Hence

By writing the integral defining K λμ(m, n) as a limit of approximating sums, we see that (A −λI)Kλμ(m, n) is defined and that it is given by the sum of two integrals, the first of which vanishes (by Cauchy’s theorem) and the second gives K λμ(m +1, n). We have thus shown that Kλμ(m, n) maps into D(A) and

Similarly

We also have

Proof. We have ([A −λI]Kλμ(m, n)y, Kλμ(m, n)y)

A similar argument shows that A ≤ μI there. Thus if we define Mλμ(m, n) to be the closure of im Kλμ(m, n) we see that A is bounded when restricted to M λμ(m, n) and

there. We let Nλμ(m, n) denote the kernel of Kλμ(m, n) so that Mλμ(m, n) and Nλμ(m, n) are the orthogonal complements of one another. So far we have not made use of the assumption that A has no point spectrum. Here is where we will use this assumption: Since

we see that if Kλμ(m +1, n)x = 0 wemusthave (A −λI)Kλμ(m, n)x = 0 which, by our assumption, implies that K λμ(m, n)x = 0. In other words, Proposition 20.2.2. The space Nλμ(m, n), and hence its orthogonal complement Mλμ(m, n) is independent of m and n. We will denote the common space M λμ(m, n) by M λμ.We have proved that A is a bounded operator when restricted to M λμ and satisfies

there. We now claim: Proposition 20.2.3.

Proof. Let Cλμ denote the rectangle of height one parallel to the real axis and cutting the real axis at the points λ and μ. Use similar notation to define the rectangles Cλν and Cνμ. Consider the integrand

and let

with similar notation for the integrals over the other two rectangles of the same

integrand. Then clearly

Also, writing z I − A = (z −ν)I + (ν I − A) we see that

Since A has no point spectrum, the closure of the image of Tλμ is the same as the closure of the image of K λμ(1, 1), namely M λμ. The proposition now follows from (20.11). If we now have a doubly infinite sequence as in our reformulation of the spectral theorem, and we set Mi ≔ M λi λi+1 we have proved the spectral theorem (in the no point spectrum case—and hence in the general case) if we show that

In view of (20.10) it is enough to prove that the closure of the limit of M−rr is all of as r →∞, or, what amounts to the same thing, if y is perpendicular to all K−rr (1, 1)x then y must be zero. Now

so we must show that if K−rr y = 0 for all r then y = 0. Now

where we may take C to be the circle of radius r centered at the origin. We also have

So so (pulling the A out from under the integral sign) we can write the above equation as

On C we have z = reiθ so z2 = r2e2iθ = r2(cos 2θ + i sin 2θ) and hence z2 −r 2 = r 2(cos 2θ −1+i sin 2θ) = 2r2(−sin2 θ +i sin θ cos θ). Now

so we see that

Since |z−1| = r−1 on C, we can bound ║gr║ by

as r →∞. Since y = Agr and A is closed (being self-adjoint) we conclude that y = 0. This concludes Lorch’s proof of the spectral theorem.

Chapter 21

Scattering Theory via Lax and Phillips I want to spend a few chapters on “scattering theory.” The basic text is Volume III of Reed and Simon [17] devoted to this topic and I really have no business getting into this subject, but . . .. The purpose of this chapter is to give an introduction to the ideas of Lax and Phillips, which is contained in their beautiful book [11]. 21.1

Notation

Throughout this chapter K will denote a Hilbert space and a strongly continuous semi-group of contractions defined on K which tends strongly to 0 as t →∞ in the sense that

21.1.1

Examples

Translation-truncation. Let N be some Hilbert space and consider the Hilbert space

Let Tt denote the one parameter unitary group of right translations:

and let P denote the operator of multiplication by 1(−∞,0] so P is projection onto the subspace G consisting of the f which are supported on (−∞, 0]. We claim that

is a semi-group acting on G satisfying our condition (21.1) of tending strongly to 0 as t →∞: The operator PTt is a strongly continuous contraction since it is a unitary operator on L2(ℝ,N) followed by a projection. Also

tends strongly to zero as t →+∞. We must check the semi-group property: Clearly PT0 = Id on G. We have

where

so

for s ≥ 0. Hence

since

Axiomatizing the above: Incoming representations. The last argument is quite formal. We can axiomatize it as follows: Let H be a Hilbert space, and a strongly continuous one parameter group of unitary operators on H. A closed subspace D ⊂ H is called incoming with respect to U if

Then S (t) := PDU(t) is a strongly continuous semi-group. Let PD: H → D denote orthogonal projection. The preceding argument goes over unchanged to show that S defined by

is a strongly continuous semi-group. Repeat of the above argument in general: The operator S(t) is clearly bounded and depends strongly continuously on t. For s and t ≥ 0 we have

where

But g ∈ D⊥ ⇒U(s)g ∈ D⊥ for s ≥ 0 since U(s)g = U(−s)∗g and

by (21.2) which says that U(−s)ψ ∈ D for s ≥ 0. S (t) converges strongly to zero as t →∞. We know that

and want to prove that

First, observe that (21.2) implies that if s < t then D ⊃ U(s −t)D so

and since U(−s)D⊥ = [U(−s)D]⊥ we get

We claim that

is dense in H. If not, there is a 0 = h ∈ H such that h ∈ [U(−t)D⊥]⊥ for all t > 0 which says that U(t)h ⊥ D⊥ for all t > 0 or U(t)h ∈ D for all t > 0 or

contradicting

Therefore, if f ∈ D and > 0 we can find g ⊥ D and an s > 0 so that

or

Since g ⊥ D we have PD[U(s) f − g] = PDU(s) f and hence

proving that PDU(s) tends strongly to zero. Comments about the axioms. Conditions (21.2)–(21.4) arise in several seemingly unrelated areas of mathematics. In scattering theory—either classical where the governing equation is the wave equation, or quantum mechanical where the governing equation is the Schrödinger equation—one can imagine a situation where an “obstacle” or a “potential” is limited in space, and that for any solution of the evolution equation perpendicular to the eigenvectors, very little energy remains in the regions near the obstacle as t → −∞ or as t → +∞. In other words, the obstacle (or potential) has very little influence on the solution of the equation when |t| is large.

We can therefore imagine that for t ≪ 0 the solution behaves as if it were a solution of the “free” equation, one with no obstacle or potential present. Thus the meaning of the space D in this case is that it represents the subspace of the space of solutions which have not yet had a chance to interact with the obstacle. The meaning of the conditions should be fairly obvious. In data transmission. We are all familiar with the way an image used to come in over the internet: first blurry and then with an increasing level of detail. In wavelet theory there is a concept of “multiresolution analysis,” where the operators U provide an increasing level of detail. Martingales.We can allow for the possibility of more general concepts of “information”; for example, in martingale theory where the spaces U(t)D represent the space of random variables available based on knowledge at time ≤ t. In the data transmission example, it is more natural to allow t to range over the integers, rather than over the real numbers. In fact, Lax and Phillips start by dealing with the discrete case which is simpler, and then pass to the continuous case via the Cayley transform. In the martingale example, we might want to dispense with U (t) altogether, and just deal with an increasing family of subspaces. In the scattering theory example, we want to believe that at large future times the “obstacle” has little effect and so there should be both an “incoming space” describing the situation long before the interaction with the obstacle, and also an “outgoing space” reflecting behavior long after the interaction with the obstacle. The residual behavior (i.e., the effect of the obstacle) is of interest. For example, in elementary particle physics, this might be observed as a blip in the scattering cross-section describing a particle of a very short lifetime. See the very elementary discussion of the blip arising in the Breit-Wigner formula below. 21.2

Incoming and Outgoing Subspaces

So let be a strongly continuous one parameter unitary group on a Hilbert space H, let D− be an incoming subspace for U and let D+ be an outgoing subspace (i.e., incoming for Suppose that

and let

Let

Let

Claim: Z(t): K →K.

Proof. Since P+ occurs as the leftmost factor in the definition of Z, the image of Z (t) is contained in We must show that

since Z(t)x = P+U(t)x as the conditions for incoming, and so

Now U(−t) : D− →D− for t ≥ 0 is one of

So

Since

the projection P+ is the identity on D−, in particular

and hence, since P+ is self-adjoint,

Thus

as required.

Change of notation: We will now use Z (t) to denote the restriction of P+U (t)P − to K.

is a semi-group. We claim that

is a semi-group. Indeed, we have

since [P+x − x] ∈ D+ and for t ≥ 0. Also Z(t) = P+U(t) on K since P− is the identity on K. Therefore, we may drop the P− on the right when restricting to K and we have

proving that Z is a semi-group. We now show that Z tends strongly to 0 as t →∞. For any x ∈ H and any > 0 we can find a T > 0 and a y ∈ D+ such that

since ᑌt T

and hence

We have proved that Z is a strongly contractive semi-group on K which tends strongly to zero, i.e., that (21.1) holds. 21.3

Breit-Wigner

The example in this section will be of primary importance to us in computations and will also motivate the Lax-Phillips representation theorem to be stated and proved in the next section. Suppose that K is one-dimensional, and that

for d ∈ K where

This is obviously a strongly contractive semi-group in our sense. Consider the space L2(R,N) where N is a copy of K but with the scalar product whose norm is

Let

Then

so the map

is an isometry. Also P: L2 → L2((−∞, 0]) satisfies

so

This is an example of the representation theorem in the next section. The Breit-Wigner function. If we take the Fourier transform of fd we obtain the function

whose norm as a function of σ is proportional to the Breit-Wigner function

It is this “bump” appearing in graph of a scattering experiment which signifies a “resonance”, i.e., an “unstable particle” whose lifetime is inversely proportional to the width of the bump. 21.4

Strongly Contractive Semi-groups

Let be a strongly contractive semi-group on a Hilbert space K. We want to prove that the pair K, S is isomorphic to a restriction of our original truncation example: Theorem 21.4.1 (Lax-Phillips). There exists a Hilbert space N and an isometric map ℝ of K onto a subspace of P (L2(R,N)) such that

for all t ≥ 0. Here, and for the rest of this chapter, P denotes, as before, the operator of multiplication by 1(−∞,0], i.e., projection onto the functions supported on the negative real line. Let B be the infinitesimal generator of S, and let D(B) denote the domain of B. The sesquilinear form

is non-negative definite since, according to the Lumer-Phillips theorem on contracting semi-groups, B satisfies the Lumer-Phillips theorem

Dividing out by the null vectors and completing gives us a Hilbert space N whose scalar product we will denote by (,)N. If k ∈ D(B), so is S(t)k for every t ≥ 0. Let us define

where [S(t)k] denotes the element of N corresponding to S(t)k. For simplicity of notation we will drop the brackets and simply write

and think of fk as a map from (−∞, 0] to N. We have

Integrating this from 0 to r gives

the norms on the right being those of K. By hypothesis, the second term on the right tends to zero as r →∞. This shows that the map

is an isometry of D(B) into L2((−∞, 0],N), and since D(B) is dense in K, we conclude that it extends to an isometry of K with a subspace of P (L2(R,N)). Also

is given by

Thus R(K) is an invariant subspace of P (L2(ℝ,N)) and the intertwining equation of the theorem holds. We can strengthen the conclusion of the theorem for elements of D(B): Proposition 21.4.1. If k ∈ D(B) then fk is continuous in the N norm for t ≤ 0. Proof. For s, t > 0 we have

by the Cauchy-Schwarz inequality. Since S is strongly continuous the result follows. Let us apply this construction to the semi-group associated to an incoming space D for a unitary group U on a Hilbert space H. Let d ∈ D and fd = Rd as above. We know that U(−r)d ∈ D for r > 0 by (21.2). Notice also that

for d ∈ D. Then for t ≤ −r we have, by definition,

and so by the Lax-Phillips theorem,

Since U (−r) is unitary, we have equality throughout which implies that

We have thus proved that if

then

21.5

The Sinai Representation Theorem

21.5.1

The Sinai Representation Theorem

Theorem 21.5.1. If D is an incoming subspace for a unitary one parameter group, acting on a Hilbert space H then there is a Hilbert space N, a unitary isomorphism

such that

and

where P is projection onto the subspace consisting of functions which vanish on (0,∞] almost everywhere. Proof. We apply the results of the last section. For each d ∈ D we have obtained a function fd ∈ L2((−∞, 0],N) and we extend fd to all of ℝ by setting fd (s) = 0 for s > 0. We have thus defined an isometric map R from D onto a subspace of L2(R,N). Now extend R to the space U (r)D by setting

Equation

assures us that this definition is consistent in that if d is such that U (r)d ∈ D then this new definition agrees with the old one. We have thus extended the map R to as an isometry satisfying

Since

is dense in H the map R extends to all of H. Also by construction

where P is projection onto the space of functions supported in (−∞, 0] as in the statement of the theorem. We must still show that R is surjective. For this it is enough to show that we can approximate any simple function with values in N by an element of the image of R. Recall that the elements of the domain of B, the infinitesimal generator of PDU(t), are dense in N, and for d ∈ D(B) the function fd is continuous, satisfies f (t) = 0 for t > 0, and f (0) = n where n is the image of d in N. Hence

is mapped by R into a function which is approximately equal to n on [0, δ] and zero elsewhere. Since the image of R is translation invariant, we see that we can approximate any simple function by an element of the image of R, and since R is an isometry, the image of R must be all of L2(R,N). 21.5.2

The Stone-von Neumann Theorem

In this section we show that the Sinai representation theorem implies a version (for n = 1) of the celebrated Stone-von Neumann theorem: Theorem 21.5.2. Let {U(t)} be a one parameter group of unitary operators, and let B be a self-adjoint operator on a Hilbert space H. Suppose that

Then we can find a unitary isomorphism ℝ of H with L2(R,N) such that

and

where mx is multiplication by the independent variable x. Remark. If i A denotes the infinitesimal generator of U, then differentiating (21.6) with respect to t and setting t = 0 gives

which is a version of the Heisenberg commutation relations. So (21.6) is a “partially integrated” version of these commutation relations, and the theorem asserts that (21.6) determines the form of U and B up to the possible “multiplicity” given by the dimension of N. Proof. By the spectral theorem, write

where {Eλ} is the spectral resolution of B, and so we obtain the spectral resolutions

and

by a change of variables. We thus obtain

Remember that E λ is orthogonal projection onto the subspace associated to (−∞, λ] by the spectral measure associated to B. Let D denote the image of E0. Then the preceding equation says that U(t)D is the image of the projection E t. The standard properties of the spectral measure—that the image of E t increase with t, tend to the whole space as t →∞and tend to {0} as t →−∞—are exactly the conditions that D be incoming for U (t). Hence the Sinai representation theorem is equivalent to the Stone-von Neumann theorem in the above form. Historically, Sinai proved his representation theorem from the Stone-von Neumann theorem. Here, following Lax and Phillips, we are proceeding in the reverse direction.

I refer the reader to the book by Lax and Phillips for the full development of their theory. 21.6

The Stone-von Neumann Theorem

In this section I will outline a discussion of the above famous theorem, referring to the book [12] for details and proofs. As mentioned in the preceding section, this theorem asserts the uniqueness of the “representation of the Heisenberg commutation relations in Weyl form.” This theorem was conjectured by Hermann Weyl soon after Heisenberg formulated his commutation relations, but was first proved independently by Stone and von Neumann in the early 1930s. It is more or less taken for granted in the physics texts and used as one of the foundation stones of quantum mechanics. 21.6.1

The Heisenberg Algebra and Group

Let V be a finite dimensional vector space over the real numbers. A symplectic structure on V consists of an antisymmetric bilinear form

which is non-degenerate. A vector space equipped with a symplectic structure is called a symplectic vector space. From a symplectic vector space V we make h ≔ V ⊕R into a Lie algebra by defining:

for X, Y ∈ V, where E = 1 ∈ ℝ and

The Lie algebra h is called the Heisenberg algebra. It is a nilpotent Lie algebra. In fact, the Lie bracket of any three elements is zero. We will let N denote the simply connected Lie group with this Lie algebra. We may identify the 2n +1-dimensional vector space V +R with N via the exponential map, and with this identification the multiplication law on N reads

Let dv be the Euclidean (Lebesgue) measure on V. Then the measure dvdt is invariant under left and right multiplication. So the group N is unimodular. The group N is called the Heisenberg group. For those who are unfamiliar with the notion of the exponential map for Lie algebras and Lie groups, just start with (21.7) as a definition of multiplication, where exp is just a weird symbol. 21.6.2

The Stone-von Neumann Theorem

The Stone-von Neumann theorem asserts that there is a unique irreducible unitary representation ρ of N such that

We check that (in terms of a suitable basis) the (infinitesimal) representation of h corresponding to ρ are the Heisenberg commutation relations. For details and proofs see [12].

Chapter 22

Huygens’ Principle We will need Huygens’ principle in a later discussion of quantum scattering, so a brief summary of properties of the wave equation that lead up to it might be appropriate here. 22.1

Introduction

We want to study various properties of the wave equation

in n +1 dimensions. In practical applications, the wave equation has the form

where c is a velocity. So (22.1) represents (22.2) in an appropriate choice of coordinates. In vague terms, we want to establish the following main points: We will let

so that the “Cauchy” or “initial value problem” consists of looking for a solution to (22.1) with the above initial conditions, where f and g are given. We are aiming to prove: •

Existence and uniqueness. The solution to the initial value problem exists

and is unique. Finite propagation velocity. The u(t0, x0), t0 ≥ 0 depends only on the values of f and g in the ball Huygens’ principle. If n ≥ 3 is odd, the value of u at (t0, x0), t0 ≥ 0 depends only on the values of f and g and some of their derivatives on the sphere

• • •

Propagation of the wave front. For all dimensions, even or odd, the singularities of u depend only on the singularities of f and g on the sphere

•

Conservation of energy. The “energy”

is constant in t. 22.2

d’Alembert and Duhamel

Huygens lived in the 17th century so our subject goes back to then. We will begin with the work of d’Alembert (18th century) and Duhamel (19th century). So we will look at the wave equation in one dimension and put the c back in: The initial value problem for the wave equation is

Then

is the d’Alembert solution of the one-dimensional wave equation. Propagation velocity. It is immediate to check that

is a solution of the wave equation with the given initial conditions. The interval

[xt −ct1, x1 +ct1], t > 0 of the x-axis is the domain of dependence of the point (x1, t1). The reason for this name is that (22.5) indicates that u(x1.t1) depends only on the values of ϕ taken at the ends of this interval and the values of ψ at all points of this interval. The region D for t > 0, x −ct ≤ x1, and x +ct ≥ x1 is the domain of influence of x1. The general solution of the wave equation. Temporarily ignoring initial values, the general solution of the wave equation is of the form

i.e., the sum of a “wave” moving unchanged to the right with velocity c and one moving to the left. Oscillatory solutions. If we assume a separated form

then (22.4) becomes

The solution is

Thus

So by (22.6), for waves with time dependence exp(−iωt) • •

exp(ikx) characterizes a wave travelling to the right (increasing x) and exp(−ikx) characterizes a wave travelling to the left (decreasing x).

Outgoing and incoming solutions in one dimension. Getting back to the d’Alembert solution, let us define

so that the d’Alembert solution is given as

u+(x, t) is a function of x −ct for x > 0 and a function of x +ct for x < 0. So with respect to the origin it is “outgoing” and similarly u− is “incoming.” Using functional analysis. The operator is (or can be considered as) a nonnegative self-adjoint operator on L2(ℝ). So we can consider, more generally, the ordinary differential equation

where A is a non-negative self-adjoint operator on a Hilbert space. So via the functional calculus this equation with initial conditions u(·, 0) = ϕ, ut (·, 0) = ψ has the solution

Duhamel’s principle. Let us now consider the inhomogeneous equation

with initial conditions u(·, 0) = ϕ, ut (·, 0) = ψ. We can write such a solution as a sum u = v +w of where v is a solution of the homogenous equation with the given initial conditions and a solution w of the inhomogeneous equation with zero initial conditions. To solve the equation for w consider the operator

Let

Then w(0) = 0 and

and

so the general solutions of (22.10) is

This is (a case of) Duhamel’s principle. N-dimensions. We will take c = 1 and go back to our choice of f and g to denote the initial conditions. We will also eventually move from the 19th to the 20th century. In a while, we will find that the key difference between even and odd dimensions lies in the inversion formula for the Radon transform. 22.3 22.3.1

Fourier Analysis Using the Fourier Transform to Solve the Wave Equation

We will discuss the issues raised above from several points of view: Assume that f, g ∈ S and look for u such that u(t, ·) ∈ S for all t. Applying the Fourier transform converts the wave equation (22.1) into

which is an ordinary differential for each fixed ξ whose solution for the given initial data is:

cos(θ) is a function of θ2 and so is sin(θ)/θ so the right-hand side is smooth as a function of ξ and t. The uniqueness theorem for ordinary differential equations gives the uniqueness of the above solution, at least if we demand that the solution belong to S for any fixed t. 22.3.2

Conservation of Energy at Each Frequency

The Fourier transform gives the expression of the energy in terms of ξ as

We will prove much more than the assertion that E (t) is constant. We will prove that the integrand in (22.14) is constant. Indeed, from our solution (22.13) we obtain, by differentiating and by multiplying by ║ξ║, the equations

We can write this as the matrix equation:

This shows that

and so is independent of t. 22.3.3

Distributional Solutions

Some Hilbert spaces of generalized functions. Let space Hs to consist of all u ∈ S'(the dual space of S) which satisfy

with norm

Define the

Thus H0 = L2. For example, since the Fourier transform of the δ function is a constant, and since ⟨ξ⟩s is integrable if and only if 2s < −n we see that δ ∈ Hs if and only if On the other hand, the Fourier transform of the constant one is a multiple of the δ function, and so does not belong to Hs for any s. So s describes the degree of smoothness, but to belong to Hs a distribution has to “vanish at infinity” sufficiently fast. 22.4

The Radon Transform

Parameterize the hyperplanes in Rn in a two to one fashion by the pairs (s,ω) ∈ ℝ × Sn−1. Here the hyperplane

is the hyperplane orthogonal to the direction ω at signed distance s from the origin, where the sign of s is chosen so that sω belongs to the hyperplane. The pairs (s,ω) and (−s,−ω) parameterize the same hyperplane. On the hyperplane H we will use dHz to denote the measure induced by the Euclidean metric. 22.4.1

Definition of the Radon Transform

The Radon transform R maps functions on ℝn (initially elements of functions on ℝ × Sn−1 by averaging over hyperplanes:

to

Clearly

Relation to the Fourier transform. The Radon transform is related to the Fourier transform as follows:

In other words,

22.4.2

Using the Fourier Inversion Formula for the Wave Equation

The Fourier inversion formula, using polar coordinates, says

Putting in our relation between the Fourier transform and the Radon transform gives

In this integral, r is positive so we may write |r| instead of r. If we then make the replacements we can rewrite the last integral as an integral over all r. So we get

We can then apply the Fourier inversion formula. For any function of the variable s define the operator |Ds|n−1 by

where m|σ|n−1 denotes multiplication by |σ|n−1. If n is odd, , the usual differential operator. So we get the inversion formula for the Radon transform:

Applying the inversion formula to Δu (which at the Fourier transform level means multiplication by −║ξ║2) shows that

so the Radon transform takes the n-dimensional Laplacian into the onedimensional Laplacian. We can use this fact to write a formula for the solution and verify Huygens principle in odd dimensions > 1. 22.4.3

Huygens’ Principle in Odd Dimensions > 1

Recall that for n = 1 the solution to the initial value problem for the wave equation is given by d’Alembert’s formula

Notice that the dependence on g extends to the entire interval [x −t, x +t] but the values of ∂xu and ∂t u depend only on the values of f' and g at x ±t. Now let us consider the wave equation in odd dimensions greater than one. Applying the Radon transform in the x variables to u and to the initial conditions, and using we get

If we now apply the inverse Radon transform, we see, because of the presence of the in the inversion formula, that u(x, t) depends only on the values of f, g and their derivatives on the sphere This is Huygens’ principle!

Chapter 23

Some Quantum Mechanical Scattering Theory 23.1 23.1.1

Introduction Notation

will denote a Hilbert space. H0 and H will denote self-adjoint operators on . The corresponding one parameter groups are

(We are taking The idea is to compare U(t) with U0(t) for t → ±∞. We set

23.1.2

The Spaces M + and M −

We let M± denote the set of all

such that the limits

exist. The operators W± are called the wave operators.

Theorem 23.1.1. M± are closed subspaces of . Proof. Let un ∈ M + with un →u. Now

(since W(t) is unitary) for any k, t1 and t2. Choose k so large that

and then t1 and t2 so large that

Then

Similarly for M−.

The spaces R±. The operators W± are defined on the spaces M± and are isometries. We define

Theorem 23.1.2. R± are closed subspaces of . Proof. Let fk ∈ R+ with fk → f . By assumption, fk = W+uk for uk ∈ M+. Since W+ is an isometry, the uk converge, and since M+ is closed, uk →u ∈ M+, and

so f = W+u. Similarly for R−. 23.1.3

Scattering States and the Scattering Operator

Let ψ(0) and ψ−(0) be elements of and set

If

then we say that ψ−(t) is an incoming asymptotic state for ψ(t). If ψ+(0) is such that ψ+(t) ≔ U0(t)ψ+(0) satisfies

then we say that ψ+(t) is an outgoing asymptotic state for ψ(t). Rephrasing the definition of asymptotic states.

so this approaches 0 as t →+∞ if and only if ψ+(0) ∈ M+ and ψ(0) = W+ψ+(0). Similarly for incoming asymptotic states. So a state ψ(t) has an incoming asymptotic state if and only if ψ(0) belongs to R− and has an outgoing asymptotic state if and only if ψ(0) belongs to R+. Scattering states. A state ψ(t) = U (t)ψ(0) is called a scattering state if it possesses an incoming and an outgoing asymptotic state. Theorem 23.1.3. ψ(t) is a scattering state if and only if ψ(0) ∈ R+ ∩ R−. 23.2

The Scattering Operator

Since the wave operators W± are isometries, they are injective. So if ψ(t) is a scattering state, the ψ±(0) are uniquely determined by the equation ψ(0) = W±ψ ±(0). So we may write

The operator

is called the scattering operator. Clearly its domain, D(S), consists of those u ∈ M− such that W−u ∈ R+ and its range R(S) consists of those f ∈ M+ such that W+ f ∈ R−.

So if R− ⊂ R+ then D(S) = M−. Conversely, if R− ⊂ R+ then there is a u ∈ M− such that W−u is not in R+ so is not defined. Similarly, the range R(S) consists of those f ∈ M+ such that W+ f ∈ R−. So if R+ ⊂ R− then R(S) ⊃ M+ and otherwise not. So we have proved Theorem 23.2.1. • D(S) = M− ⇔ R− ⊂ R+, • R(S) = M+ ⇔ R+ ⊂ R−, and • S is a unitary map from M− to M+ if and only if R− = R+. Intuitive meaning. Suppose that H0 is the free Hamiltonian on ℝN and that H = H0 + V where V is a Kato potential, so that D(H) = D(H0). We know from Ruelle’s theorem that the subspace of corresponding to the discrete spectrum of H consists of the “bound states” which “remain in a bounded region of space” for all time, while the subspace c ⊂ corresponding to the continuous spectrum correspond to the “scattering states” which fly off to infinity. We expect that for large t, “when the state is far away from the origin,” the behavior of such ψ(t) should be like a solution of the free Hamiltonian, and similarly for very negative t, at least if V vanishes sufficiently fast at infinity. So we want to look for conditions on V such that R+ = c = R− and M+ = M− = . We will begin by deriving some more abstract conditions in general on the wave operators and the scattering operator, and then getting more detailed results by using the resolvent. 23.3 23.3.1

Properties of the Wave and Scattering Operators Invariance Properties of M± and R ±

Proposition 23.3.1. For any t, U0(t)M± ⊂ M± and U(t)R± ⊂ R±. Proof. Suppose that W−u = f so that

Then

This shows that U0(t)u ∈ M− and W−U0(t)u = U(t) f so U(t)R− ⊂ R−. Similarly for M+ and R±. In the course of the proof we have shown that

Orthogonal projections onto M±. Let P± denote orthogonal projection onto M±. So U0(t)P± = P±U0(t) for all t. By Stone’s theorem u ∈ D(H0) if and only if the limit

exists, and then this limit equals −i H0u. It then follows that P±D(H0) ⊂ D(H0) and if u ∈ D(H0) then

A similar result holds for H and the orthogonal projections onto R±. A sufficient condition to be an element of M±. Assume that u is such that U0(t)u ∈ D(H0) ∩ D(H) ∀ t > a. For example, this will hold for all u ∈ D(H0) if D(H) = D(H0) as is the case for the free Hamiltonian plus a Kato potential. Suppose also that

∞a ║W'(t)║dt is finite. Then for t2 ≥ t1 > a

So W(t)u converges as t →∞. 23.3.2

Cook’s Lemma

Now W(t) = U(−t)U0(t) so

We have proved: Theorem 23.3.1. If U0(t)u ∈ D(H0) ∩ D(H) ∀ t > a and

then u ∈ M+. A similar result holds for M −. L2 potentials on ℝ3. Theorem 23.3.2. Let H0 be the free Hamiltonian on ℝ3 and H = H0 + V where V ∈ L2(ℝ3). Then M± = = L2(ℝ3). Proof. For u we have the explicit solution

so

Thus

and the integral in Cook’s lemma converges. Since ± are closed, we see that M± = .

is dense in and the M

Potentials on ℝ1. The above argument clearly works for dimensions ≥ 3, but not for one dimension. If we put the condition on V that |x|r | V (x)| be bounded for some r > 1 in addition to V belonging to L2, a modification of the above argument will show that the integral in Cook’s lemma converges and so M± = . The details are straightforward but somewhat technical so I will skip them. 23.3.3

The Chain Rule

We will use the case where there are three Hamiltonians H1, H2, and H3 and none of them have bound states. We will also use Ω± instead of W± to conform with certain texts that will be used later. Proposition 23.3.2. If Ω±(H1, H2) and±(H2, H3) exist then±(H1, H3) exists and

Proof.

Both of these last two terms go to zero. Why not the uniform topology? In defining the wave operators we took the strong limit of W(t) = U(−t)U0(t) as t →±∞. There is no such limit in the uniform topology unless U = U0. Indeed, suppose that

Then

Hence by the uniqueness of the limit we must have

Hence

23.4 23.4.1

“Time Independent Scattering Theory” Expressing the Wave Operators in Terms of the Resolvents

Recalling some formulas for the resolvent. Let H be a self-adjoint operator on a Hilbert space . Its resolvent

is given, for any complex number z with positive imaginary part by

This follows from the spectral theorem and the formula

Similarly, if the imaginary part of z is negative, then

Of course, both of these formulas are special cases of the general fact that the resolvent is the Laplace transform of the semi-group. If we write z = x +iy, y > 0 (23.2) becomes

This is

the inverse Fourier transform of the function

where χ[0,∞) is the indicator function of the positive real axis, i.e., the Heaviside function. So if denotes the inverse Fourier transform, we have

Similarly, if R0(z) denotes the resolvent of H0 we have

By Parseval’s theorem

Since (U0(t)u,U(t)v) = (W(t)u, v) this becomes

Abel’s lemma. If ϕ is a bounded, continuous function of t with |ϕ(t)| ≤ M and ϕ(t) → 0 as t →∞then writing ϕ = ϕ0 + ϕ1 where ϕ0 is supported on an interval [0, R] and sup |ϕ1(t)| ≤ we have

So

So if ψ is a bounded, continuous function of t with |ψ(t)| ≤ M and ψ(t) →c as t →∞then

Applied to

we conclude that if u ∈ M+ then

Similarly, if u ∈ M− then

Notice that time has disappeared from these formulas, which is the reason for the name “time independent” scattering theory. 23.4.2

A Perturbation Series for the Wave Operators

In the formulas

it is frequently the case that we know an explicit formula for R0. For example, we wrote down such a formula for the free Hamiltonian on ℝ3 along the negative real axis. But we may not have a reasonable formula for R(z). We will show that there is a series expansion for R in terms of R0 and H − H0 (which may or may not converge). This series will have as its first term R0. So let us first consider the trivial case where H = H0 so W± = I, the identity operator. It then follows from the above formula that

Recall: The second resolvent identity. Theorem 23.4.1. Let H and H0 be two operators with the same dense domain D. For any z which belongs to the resolvent set of both we have

A “series” for R(z). Let V ≔ H − H0 and write the second resolvent identity as

and iterate to obtain

So if (R0(z)V)n+1R(z) →0 we get a series expansion for R(z). If we take the first two terms we get the Born approximation. A mnemonic. A mnemonic for the “series”

is that for numbers v = h −h0 we have

so

23.5

Meromorphic Continuation of the Resolvents

I want to study the resolvent for the free Hamiltonian and the Hamiltonian with potential a bit more closely: Recall that in three dimensions we proved that if we define G0(λ) = −R(H0, λ2) for λ > 0, G0 is given by convolution with

As an operator on L2 this is only defined for But if f is bounded and of compact support, then the convolution with g0(λ) is defined for all values of λ and defines a map

depending holomorphically on λ. I want to extend this to odd dimensions≥ 3. For this it is convenient to use the wave equation.

The free resolvent. Consider the operator (defined by the functional calculus)

By ordinary calculus we have is a solution of the wave equation with initial conditions u(0) = f, u(0) = 0. Integration by parts shows that

Integration by parts. Indeed, consider

(We are interested in

Then

Integrating by parts once again gives −I −iλF. Since

we obtain

as desired. Using Huygens’ principle in odd dimensions.

I claim that this operator makes sense for all values of λ as a map Indeed let ρ be a smooth function of compact support which is identically one on the support of f. I want to show that

is well defined for all values of λ where g0 is given by the right-hand side of the above expression. If the support of ρ is contained in the ball B(R) of radius R about the origin, then Huygens’ principle tells me that the support of lies outside the ball of radius t and inside the ball of radius R + t. So if t > R we can replace the by the finite integral 23.6

The Lippmann-Schwinger Equation—A Reference

The key mathematical ideas explaining the Lippmann-Schwinger equation and its application to scattering theory are due to Teruo Ikebe [9]. This material is beautifully explained in Vol. 3 of Reed and Simon, and I find, reviewing my notes, that I have nothing to add to (or subtract from) their treatment so I refer the reader to them. .

Chapter 24

The Groenewold-van Hove Theorem This is a “no-go” theorem which asserts that the “standard quantization” of polynomials of degree ≤ 2 cannot be extended to any larger Poisson sub-algebra of the algebra of polynomials. The provided proof is taken frommy book with Guillemin, Symplectic Techniques in Physics. I am reproducing it here due to the many typos in that section of the book. The actual proof is due to the late Paul Chernoff. 24.1

Poisson Algebra, Notation

A Poisson algebra is a commutative algebra that is also a Lie algebra, such that the Lie bracket (denoted by { , } and called the Poisson bracket) is a derivation of the multiplicative structure. The convention we will follow in this chapter is that the Poisson bracket of two functions f and g of (q, p) is

so, for example,

and

The idea of Dirac was to assign an operator to each f in such a way that the

Poisson bracket goes over into commutator bracket. The “standard quantization” assigns skew-adjoint operators to polynomials of degree ≤ 2 according to the following table: Polynomial →Corresponding operator

You can check by direct computation that this assignment takes Poisson brackets into commutator brackets. A more abstract reason why this works depends on the Stone-von Neumann theorem. The Groenewold-van Hove Theorem asserts that this cannot be extended to any Poisson sub-algebra of the algebra of all polynomials which strictly contains all polynomials of degree ≤ 2. The proof goes as follows: 1. The polynomials of degree ≤ 2 form a maximal Poisson sub-algebra of the algebra of all polynomials. 2. In particular, any such sub-algebra that strictly contains all polynomials of degree ≤ 2 contains q2 p2. 3. Express q2 p2 in two different ways as Poisson brackets of other elements leading to two different forced expressions for their “quantization.” 24.1.1 The Polynomials of Degree ≤ 2 Form a Maximal Poisson Subalgebra of the Algebra of All Polynomials The details follow. 24.1.2

The Action of sp(2) on Homogeneous Polynomials

Let sp (2) denote the Lie algebra (under Poisson bracket) of the homogenous polynomials of degree exactly 2. So a basis of sp(2) is q2, qp, p2. Let S denote the space of polynomials which are homogeneous of degree ℓ. It is clear that Sℓ is invariant under Poisson bracket by sp(2) = S2. Proposition 24.1.1. The action of sp(2) on Sℓ is irreducible. Proof. Notice that {pq, pjqk} = (j −k)pj qk and the values (j −k) are all distinct as j and k range over all non-negative integers with j +k = ℓ. So any non-zero invariant subspace must contain a monomial. But

so repeated bracketing by shows that any non-zero invariant subspace must contain Then repeated bracketing by gives all monomials. 24.1.3

Maximality of the Polynomials of Degree ≤ 2

Theorem 24.1.1. The polynomials of degree ≤ 2 form a maximal Poisson subalgebra of the algebra of all polynomials. Proof. If f is any polynomial of degree> 2, repeated bracketing by p or q would yield a cubic polynomial. Subtracting the terms of degree< 3 yields a homogenous cubic polynomial. So a sub-algebra strictly containing the quadratic polynomials must, by the proposition, contain all homogeneous cubic polynomials, in particular, q3 and p3. Since {q3, p3} = 9q2 p2, it must contain all of S4, in particular, it must contain p4. Since {q3, p4} = 12q2 p3 it must contain all of S5, etc. 24.1.4

Expressing q2 p2 in Two Different Ways as Poisson Brackets

For future use we record

and

24.2

A Change in Notation to Self-adjoint Operators

Set A(f) = i× the formulas given in the table, so as to get self-adjoint operators and so we get the table: Polynomial→ Corresponding operator

This is the usual assignment given in quantum mechanics texts and the quantization rules become [Af, Ag] = i A({f, g}). For example, if we set

and

then

i.e.,

(where we write i for the operator i I as in the table) and

so

verifying the “Heisenberg commutation relations” in this case. From the table it follows that

Also {q2, p2} = 4qp and by Leibniz’s rule and [Q, P] = i we get

So

Using these facts we now claim Proposition 24.2.1. If A is any linear map from polynomials into an associative algebra satisfying [A(f), A(g)] = i A({f, g}) and A(q2) = Q2, A(p2) = P2 then we must have

For the proof, write X ≔ A(q3). Since {q3, q} = 0 we must have [X, Q] = 0. Since {q3, p} = 3q2 we must have [X, P] = 3i Q2. But Q3 satisfies these same identities: [Q3, Q] = 0, [Q3, P] = 3i Q2 by Leibniz’s rule. So if we set

then

We also have {q3, pq} = 3q3 which implies that

Proof. We have [Y, PQ] = [Y, P]Q + P[Y, Q] = 0 so

so

i.e., X = Q3. A similar argument (or symmetry) implies that A(p3) = P3. 24.3

Proof of the Groenewold-van Hove Theorem

We use to get two different answers for A(q2 p2) which will then prove the Groenewold-van Hove theorem. The answers will differ by a constant. From we get

By Leibniz’s rule, [Q3, P3]

So

We now use [Q, P] = i to move all powers of Q to the left. For example,

Similarly,

This gives

We now need a second Proposition 24.3.1. Under the same assumptions on A we have

Proof. We have

proving the first assertion, and the second follows similarly or by symmetry. We now use and the above proposition to conclude that

There are four brackets that we compute using Leibniz’s rule and moving the powers of Q to the left using the commutation relations [Q, P] = i and [Q2, P2] = 2i (QP + PQ). Here they are:

Adding the four results above and dividing by 12i gives the answer

This differs in the constant term from

proving the Groenewold-van Hove theorem.

Chapter 25

Chernoff’s Theorem 25.1

Convergence of Semi-groups

In this section we are interested in the following type of result: We would like to know that if An is a sequence of operators generating equibounded one parameter semi-groups exp t An and An → A where A generates an equibounded semi-group exp t A then the semi-groups converge, i.e., exp t An →expt A.We will prove such a result for the case of contractions. But before we can formulate the result, we have to deal with the fact that each An comes equipped with its own domain of definition, D(An). We do not want to make the overly restrictive hypothesis that these all coincide, since in many important applications they won’t. For this purpose we recall the following definition. Let us assume that F is a Banach space and that A is an operator on F defined on a domain D(A). We say that a linear subspace D ⊂ D(A) is a core for A if the closure of A and the closure of A restricted to D are the same: This certainly implies that D(A) is contained in the closure of A|D. In the cases of interest to us D(A) is dense in F, so that every core of A is dense in F. 25.1.1

Resolvent Convergence

We begin with an important preliminary result: Proposition 25.1.1. Suppose that An and A are dissipative operators, i.e., generators of contraction semi-groups. Let D be a core of A. Suppose that for

each x ∈ D we have that x ∈ D(An) for sufficiently large n (depending on x) and that

Then for any z with Re z > 0 and for all y ∈ F

Proof. We know that R(z, An) and R(z, A) are all bounded in norm by 1/Re z. So it is enough for us to prove convergence on a dense set. Since (z I − A)D(A) = F, it follows that (z I − A)D is dense in F since A is closed. So in proving (25.2) we may assume that y = (z I − A)x with x ∈ D. Then

where, in passing from the first line to the second we are assuming that n is chosen sufficiently large that x ∈ D(An). Theorem 25.1.1. Under the hypotheses of the preceding proposition,

for each x ∈ F uniformly on every compact interval of t. Proof. Let

and set ϕ(t) = 0 for t < 0. It will be enough to prove that these F valued functions converge uniformly in t to 0, and since D is dense and since the operators entering into the definition of ϕn are uniformly bounded in n, it is enough to prove this convergence for x ∈ D which is dense. We claim that for fixed x ∈ D the functions ϕn(t) are uniformly equi-continuous. To see this observe that

for t ≥ 0 and the right-hand side is uniformly bounded in t ≥ 0 and n. So to prove that ϕn(t) converges uniformly in t to 0, it is enough to prove this fact for the convolution ϕn ρ where ρ is any smooth function of compact support, since we can choose the ρ to have small support and integral and then ϕn(t) is close to (ϕn ★ ρ)(t). Now the Fourier transform of ϕn ★ ρ is the product of their Fourier transforms: We have

Thus by the proposition

in fact uniformly in s. Hence using the Fourier inversion formula and, say, the dominated convergence theorem (for Banach space valued functions),

uniformly in t. The preceding theorem is the limit theorem that we will use in what follows. However, there is an important theorem valid in an arbitrary Fréchet space, and which does not assume that the An converge, or the existence of the limit A, but only the convergence of the resolvent at a single point z0 in the right-hand plane! In the following, F is a Fréchet space and {exp(tAn)} is a family of equibounded semi-groups which is also equibounded in n, so for every semi-norm p there is a semi-norm q and a constant K such that

where K and q are independent of t and n. I will state the theorem here, and refer

you to pp. 269–271 [28] for the proof. Theorem 25.1.2 (Trotter-Kato). Suppose that {exp(tAn)} is an equibounded family of semi-groups as above, and suppose that for some z0 with positive real part there exists an operator R(z0) such that

and

Then there exists an equibounded semi-group exp(tA) such that

and

uniformly on every compact interval of t ≥ 0. 25.2

Chernoff’s Theorem

In what follows, F is a Banach space. Eventually we will restrict attention to a Hilbert space. But we will begin with a classical theorem of Lie: 25.2.1

Lie’s Formula

Let A and B be linear operators on a finite dimensional Hilbert space. Lie’s formula says that

Let

so that

Let Tn = (exp(A/n))(exp(B/n)). We wish to show that

Proof. Notice that the constant and the linear terms in the power series expansions for Sn and Tn are the same, so

where C = C(A, B). We have the telescoping sum

so

But

and

so

This same proof works if A and B are self-adjoint operators on a Hilbert space such that A + B is self-adjoint on the intersection of their domains. For a proof see Reed-Simon, vol. I, pp. 295–296 [17]. For applications this is too restrictive. So we give a more general formulation and proof following Chernoff [2]: 25.2.2

Statement of Chernoff’s Theorem

Theorem 25.2.1 (Chernoff). Let f: [0,∞) → bounded operators on F be a continuous map with and

Let A be a dissipative operator and exp(tA) the contraction semi-group it generates. Let D be a core of A. Suppose that

Then for all y ∈ F

uniformly in any compact interval of t ≥ 0. Before proceeding to the proof of Chernoff’s theorem, we record two facts: 1. Suppose that B : F → F is a bounded operator on a Banach spacewith ║B║ ≤ 1. Then for any semi-scalar product we have

so B − I is dissipative and hence exp(t (B − I)) exists as a contraction semi-group by the Lumer-Phillips theorem. We can prove this directly since we can write

The series converges in the uniform norm and we have

2. The following inequality holds:

Here is an amusing proof: Proof.

We have proved that

So to prove

it is enough to establish the inequality

Consider the space of all sequences a = {a0, a1, . . . } with finite norm relative to scalar product

The Cauchy-Schwarz inequality applied to a with ak = |k −n| and b with bk ≡ 1

gives The second square root is one, and we recognize the sum under the first square root as the variance of the Poisson distribution with parameter n, and we know that this variance is n. 25.2.3

Proof of Chernoff’s Theorem

Proof. For fixed t > 0 let

So generates a contraction semi-group by the special case of the LumerPhillips theorem we just discussed and therefore (by change of variable), so does Cn. So Cn is the generator of a semi-group

and the hypothesis of the theorem is that Cnx → Ax for x ∈ D. Hence by the limit theorem we proved above,

for each y ∈ F uniformly on any compact interval of t. Now

so we may apply (25.5) to conclude that

The expression inside the on the right tends to Ax so the whole expression tends to zero. This proves (25.4) for all x in D. But since D is dense in F and f (t/n) and exp(t A) are bounded in norm by 1 it follows that (25.4) holds for all y ∈ F. 25.3

The Trotter Product Formula

Let A and B be the infinitesimal generators of the contraction semi-groups Pt = exp(t A) and Qt = exp(t B) on the Banach space F. Then A + B is only defined on D(A) ∩ D(B) and in general we know nothing about this intersection. However, let us assume that D (A) ∩ D(B) is sufficiently large that the closure a densely defined operator and is in fact the generator of a contraction semigroup Rt. So D ≔ D(A) ∩ D(B) is a core for Theorem 25.3.1 (Trotter). Under the above hypotheses

converges uniformly on any compact interval of t ≥ 0. Proof. Define

For x ∈ D we have

so the hypotheses of Chernoff’s theorem are satisfied. The conclusion of Chernoff’s theorem asserts (25.7). A symmetric operator on a Hilbert space is called essentially self-adjoint if its closure is self-adjoint. So a reformulation of the preceding theorem in the case of self-adjoint operators on a Hilbert space says Theorem 25.3.2. Suppose that S and T are self-adjoint operators on a Hilbert space H and suppose that S + T (defined on D(S) ∩ D(T)) is essentially selfadjoint. Then for every y ∈ H

where the convergence is uniform on any compact interval of t. 25.3.1

Commutators

An operator A on a Hilbert space is called skew-symmetric if A∗ = −A on D(A). This is the same as saying that i A is symmetric. So we call an operator skew adjoint if i A is self-adjoint. We call an operator A essentially skew adjoint if i A is essentially self-adjoint. If A and B are bounded skew adjoint operators then their Lie bracket

is well defined and again skew adjoint. In general, we can only define the Lie bracket on D(AB) ∩ D(BA) so we again must make some rather stringent hypotheses in stating the following theorem. Theorem 25.3.3. Let A and B be skew adjoint operators on a Hilbert space H and let

Suppose that the restriction of [A, B] to D is essentially skew-adjoint. Then for every y ∈ H

uniformly in any compact interval of t ≥ 0. Proof. The restriction of [A, B] to D is assumed to be essentially skew adjoint, so [A, B] itself (which has the same closure) is also essentially skew adjoint. We have

for x ∈ D with similar formulas for exp(−t A), etc. Let

Multiplying out f (t)x for x ∈ D gives a whole lot of cancellations and yields

so (25.9) is a consequence of Chernoff’s theorem with 25.3.2

Feynman “Path Integrals” from Trotter

Consider the operator

given by

Recall that the domain of H0 is taken to be those ϕ ∈ L2(ℝ3) for which the differential operator on the right, taken in the distributional sense, when applied to ϕ gives an element of L2(ℝ3). Recall that the operator H0 is called the “free Hamiltonian of non-relativistic quantum mechanics.” Recall that the Fourier transform F is a unitary isomorphism of L2(ℝ3) into L2(ℝ3) and carries H0 into multiplication by ξ2 whose domain consists of those ∈ L2(R3) such that belongs to L2(ℝ3). The operator consisting of multiplication by is clearly unitary, and provides us with a unitary one parameter group. Transferring this one parameter group back to L2(ℝ3) via the Fourier transform gives us a one parameter group of unitary transformations whose infinitesimal generator is −i H0.

Now the Fourier transform carries multiplication into convolution, and the inverse Fourier transform (in the distributional sense) Hence we can write, in a formal sense,

Here the right-hand side is to be understood as a long-winded way of writing the left-hand side which is well defined as a mathematical object. As we have seen, the right-hand side can also be regarded as an actual integral for certain classes of f, and as the L2 limit of such integrals. We shall discuss this interpretation later. Let V be a function on ℝ3. We denote the operator on L2(ℝ3) consisting of multiplication by V also by V. Suppose that V is such that H0 + V is again selfadjoint. For example, if V were continuous and of compact support this would be the case by the Kato-Rellich theorem. (Realistic “potentials” V will not be of compact support or be bounded, but nevertheless in many important cases the Kato-Rellich theorem does apply.) Then the Trotter product formula says that

We have

Hence we can write the expression under the limit sign in the Trotter product formula, when applied to f and evaluated at x0 as the following formal expression:

where

If is a piecewise differentiable curve, then the action of a particle of mass m moving along this curve is defined in classical mechanics as

where ẋ ˙is the velocity (defined at all but finitely many points). Take m = 2 and let X be the polygonal path which goes from x0 to x1, from x1 to x2, etc., each in time t/n so that the velocity is |xi − xi−1|/(t/n) on the i -th segment. Also, the integral of V (X(s)) over this segment is approximately The formal expression written above for the Trotter product formula can be thought of as an integral over polygonal paths (with step length t/n) of eiSn(X) f (X(t))dnX where Sn approximates the classical action and where dnX is a measure on this space of polygonal paths. This suggests that an intuitive way of thinking about the Trotter product formula in this context is to imagine that there is some kind of “measure” dX on the space Ωx0 of all continuous paths emanating from x0 and such that

Feynman suggested this formula in 1942 in his thesis (Trotter’s paper was in 1959). It has been the basis of numerous important calculations in physics, many of which have given rise to exciting mathematical theorems that were then proved by other means. I am unaware of any general mathematical justification of these “path integral” methods in all the forms that they are used. 25.3.3

The Feynman-Kac Formula

Mark Kac introduced an important advance in 1951 where the unitary group exp(−i t (H0 + V)) is replaced by the contraction semi-group exp(−t (H0 + V)). Then the techniques of probability theory (in particular, the existence of Wiener measure on the space of continuous paths) can be brought to bear to justify a

formula for the contractive semi-group as an integral over path space. I will state and prove an elementary version of this formula which follows directly from what we have done. The assumptions about the potential are physically unrealistic, but I choose to regard the extension to a more realistic potential as a technical issue rather than a conceptual one. Let V be a continuous real valued function of compact support. To each continuous path ω on ℝn and for each fixed time t ≥ 0 we can consider the integral

The map

is a continuous function on the space of continuous paths, and we have

for each fixed ω. Theorem 25.3.4 (The Feynman-Kac formula). Let V be a continuous real valued function of compact support on ℝn. Let

as an operator on H = L2(ℝn). Then H is self-adjoint and for every f ∈ H

where Ωx is the space of continuous paths emanating from x and dxω is the associated Wiener measure. Since we have not discussed Wiener measure, I will pass on the proof of this theorem, referring to [17] for details.

Chapter 26

Some Background Material 26.1

Borel Functions and Borel Measures

A σ-algebra on a set X is a collection of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections. The intersection of any collection of σ-algebras is again a σ-algebra. So it makes sense to talk about the “minimum” σ-algebra with a given property as it is the intersection of all σ-algebras with the given property. Definition 26.1.1. The collection of Borel subsets of ℝ(or of any topological space X) is the smallest collection of subsets of ℝ (or of X) closed under taking countable unions, under countable intersections, under complements, and containing all open and closed subsets of R. A real or complex valued function f is a Borel function if the inverse image f −1(U) is a Borel set for every open subset U of ℝ or of C. It is easy to check that every pointwise limit of Borel functions is a Borel function. More generally, every countable infimum and countable supremum of Borel functions is a Borel function, as is every countable lim inf and lim sup. In particular, the bounded pointwise limit of continuous functions is a Borel function. For example, the indicator function 1[a,b] (sometimes called the characteristic function) which equals 1 at x ∈ [a, b] and 0 otherwise is a Borel function. Definition 26.1.2. A Borel measure μ is an assignment of real (or sometimes complex) numbers μ(E) to Borel sets E which are countably additive on disjoint

unions. More generally, the characteristic function or indicator function χE or 1E of a Borel measurable subset E equals 1 on E and 0 outside E. A simple function is a finite linear combination of characteristic functions, i.e., is of the form

Its integral is defined (as expected) as

If f is a non-negative Borel function, its integral is defined as

For a general Borel function f let its positive and negative parts be defined as f+ (x) ≔ max{f (x), 0} and f−(x) ≔ max{− f (x), 0}. These are non-negative Borel functions. If either of them has a finite integral then we define

When both are finite, we way that f is integrable. 26.2 26.2.1

The Basics of the Geometry of Hilbert Space Scalar and Semi-scalar Products

Let V be a complex vector space. A rule assigning to every pair of vectors f, g ∈ V a complex number (f, g) is called a semi-scalar product if 1. (f, g) is linear in f when g is held fixed. 2.

This implies that (f, g) is anti-linear in g when f is held fixed. In other words, It also implies that (f, f) is real.

3. (f, f) ≥ 0 for all f ∈ V. If 3 is replaced by the stronger condition 4. (f, f) > 0 for all non-zero f ∈ V then we say that ( , ) is a scalar product. Examples. a. V = Cn, so an element z of V is a column vector of complex numbers:

and (z,w) is given by

b. V consists of all continuous (complex valued) functions on the real line which are periodic of period 2π and

We will denote this space by C ( ). Here the letter T stands for the onedimensional torus, i.e., the circle. We are identifying functions that are periodic with period 2π with functions which are defined on the circle R/2πZ. c. V consists of all doubly infinite sequences of complex numbers

which satisfy

Here

26.2.2

The Cauchy-Schwarz Inequality

This says that if (·,·) is a semi-scalar product then

Proof. For any real number t, condition 3 above says that (f −tg, f −tg) ≥ 0. Expanding out gives

Since , the coefficient of t in the above expression is twice the real part of (f, g). So the real quadratic form

is nowhere negative. So it cannot have distinct real roots, and hence by the b2 −4ac rule we get

or

This is useful and almost but not quite what we want. But we may apply this inequality to h = eiθ g for any θ. Then (h, h) = (g, g). Choose θ so that

where r = |(f, g)|. Then

and the preceding inequality with g replaced by h gives

and taking square roots gives (26.1).

26.2.3

The Triangle Inequality

For any semi-scalar product define

so we can write the Cauchy-Schwarz inequality as

The triangle inequality says that

Proof.

Taking square roots gives the triangle inequality (26.3). Notice that

since In particular, it follows from the triangle inequality and (26.4) that the set of vectors f such that form a subspace. So we can pass to the quotient space which then inherits a scalar product. We make use of this construction in Chapter 21 where we study the Lax-Phillips scattering theory. 26.2.4

Pre-Hilbert Spaces

Suppose we try to define the distance between two elements of V by

Notice that then d(f, f) = 0, d(f, g) = d(g, f) and for any three elements

by virtue of the triangle inequality. The only trouble with this definition is that we might have two distinct elements at zero distance, i.e., But this cannot happen if ( , ) is a scalar product, i.e., satisfies condition 4. A complex vector space V endowed with a scalar product is called a preHilbert space. 26.2.5

Normed Spaces

Let V be a complex vector space and let be a map which assigns to any f ∈ V a non-negative real number such that for all non-zero f. If satisfies the triangle inequality (26.3) and equation (26.4), it is called a norm. A vector space endowed with a norm is called a normed space. The pre-Hilbert spaces can be characterized among all normed spaces by the parallelogram law as we will discuss below. The reason for the prefix “pre” in “pre-Hilbert” is the following: The distance d defined above has all the desired properties we might expect of a distance. In particular, we can define the notion of “limit”: If fn is a sequence of elements of V, and f ∈ V we say that f is the limit of the fn and write

if, for any positive number , there is an N = N () such that

If a sequence converges to some limit f, then this limit is unique, since any limits must be at zero distance and hence equal. We say that a sequence of elements is Cauchy if for any δ > 0 there is a K = K(δ) such that

If the sequence fn has a limit, then it is Cauchy—just choose and use the triangle inequality. But it is quite possible that a Cauchy sequence has no limit. As an example of this type of phenomenon, think of the rational numberswith |r −s| as the distance. The whole point of introducing the real numbers is to guarantee that

every Cauchy sequence has a limit. So we say that a pre-Hilbert space is a Hilbert space if it is “complete” in the above sense—if every Cauchy sequence has a limit. Since the complex numbers are complete (because the real numbers are), it follows that Cn is complete, i.e., is a Hilbert space. Indeed, we can say that any finite dimensional pre-Hilbert space is a Hilbert space because it is isomorphic (as a pre-Hilbert space) to ℂn for some n. (See below when we discuss orthonormal bases.) The trouble is in the infinite dimensional case, such as the space of continuous periodic functions. This space is not complete. For example, let fn be the function which is equal to zero on equal to one on and extended linearly and from so as to be continuous and then extended so as to be periodic. (Thus on the interval the function is given by

If m ≤ n, the functions fm and fn agree outside two intervals of length and on these intervals | fm(x)− fn(x)| ≤ 1. So

showing that the sequence {fn} is Cauchy in the norm as in case b above.

But the limit would have to equal zero on (−π, 0) and equal to one on (0, π) and

so be discontinuous at the origin and at π. Thus the space of continuous periodic functions is not a Hilbert space, only a pre-Hilbert space. 26.2.6

Completion

Just as we complete the rationals to get the real numbers, we may complete any metric space to get a complete metric space. The completion of a normed vector space will be a complete normed vector space, which is called a Banach space. From the parallelogram law discussed below, it will follow that the completion of a pre-Hilbert space is a Hilbert space. For example, we define the Hilbert space L2( ) to be the completion of the space C ( ) of continuous periodic functions under the norm coming from scalar product introduced above 26.2.7

The Pythagorean Theorem

Let V be a pre-Hilbert space. We have

So

This is the Pythagorian theorem. We make the definition

and say that f is perpendicular to g or that f is orthogonal to g. Notice that this is a stronger condition than the condition for the Pythagorean theorem, the righthand condition in (26.5). For example, If ui is some finite collection of mutually orthogonal vectors, then so are zi ui where the zi are any complex numbers. So if

then by the Pythagorean theorem

In particular, if the ui ≠ 0, then u = 0 ⇒ zi = 0 for all i. This shows that any set of mutually orthogonal (non-zero) vectors is linearly independent. Notice that the set of functions

is an orthonormal set in the space of continuous periodic functions in that not only are they mutually orthogonal, but each has norm one. 26.2.8

The Theorem of Apollonius

Adding the equations

gives

This is known as the parallelogram law. It is the algebraic expression of the theorem of Apollonius which asserts that the sum of the areas of the squares on the sides of a parallelogram equals the sum of the areas of the squares on the diagonals. If we subtract (26.7) from (26.6) we get

Now (if, g) = i (f, g) and Re {i (f, g)} = −Im(f, g) so

so

If we now complete a pre-Hilbert space, the right-hand side of this equation is defined on the completion, and is a continuous function there. It therefore follows that the scalar product extends to the completion and, by continuity, satisfies all the axioms for a scalar product, plus the completeness condition for the associated norm. In other words, the completion of a pre-Hilbert space is a Hilbert space. A theorem of Jordan and von Neumann (which we will not prove here) is essentially a converse to the theorem of Apollonius. It says that if is a norm on a (complex) vector space V, which satisfies (26.8), then V is in fact a preHilbert space with The scalar product is then given by (26.10). 26.2.9

Orthogonal Complements

We continue with the assumption that V is pre-Hilbert space. If A and B are two subsets of V,wewrite A ⊥ B if u ∈ A and v ∈ B ⇒ u ⊥ v, in other words if every element of A is perpendicular to every element of B. Similarly, we will write v ⊥ A if the element v is perpendicular to all elements of A. Finally, we will write A⊥ for the set of all v which satisfy v ⊥ A. Notice that A⊥ is always a linear subspace of V, for any A. 26.2.10

The Problem of Orthogonal Projection

Now let M be a (linear) subspace of V. Let v be some element of V, not necessarily belonging to M. We want to investigate the problem of finding a w ∈ M such that (v −w) ⊥ M. Of course, if v ∈ M then the only choice is to take w = v. So the interesting problem is when v ∉ M. Suppose that such a w exists, and let x be any (other) point of M. Then by the Pythagorean theorem,

since (v −w) ⊥ M and (w− x) ∈ M. So

and this inequality is strict if x ≠ w. In words: if we can find a w ∈ M such that (v −w) ⊥ M then w is the unique solution of the problem of finding the point in M which is closest to v. Conversely, suppose we found a w ∈ M which has this minimization property, and let x be any element of M. Then for any real number t we have

Since the minimum of this quadratic polynomial in t occurring on the right is achieved at t = 0, we conclude (by differentiating with respect to t and setting t = 0, for example) that

By our usual trick of replacing x by eiθ x we conclude that

Since this holds for all x ∈ M, we conclude that (v −w) ⊥ M. So to find w we search for the minimum of Now and is some finite number for any x ∈ M. So there will be some real number m such that and such that no strictly larger real number will have this property. (m is known as the “greatest lower bound” of the values So we can find a sequence of vectors xn ∈ M such that

We claim that the xn form a Cauchy sequence. Indeed,

and by the parallelogram law this equals

Now the expression in parentheses converges to 2m2. The last term on the right is

Since

so

we conclude that

for i and j large enough that sequence xn is Cauchy.

This proves that the

The essential role of the completeness of M. Here is the crux of the matter: If M is complete, then we can conclude that the xn converge to a limit w which is then the unique element in M such that (v −w) ⊥ M. It is at this point that completeness plays such an important role. Put another way, we can say that if M is a subspace of V which is complete (under the scalar product ( , ) restricted to M) then we have the orthogonal direct sum decomposition

which says that every element of V can be uniquely decomposed into the sum of an element of M and a vector perpendicular to M. For example, if M is the one-dimensional subspace consisting of all (complex) multiples of a non-zero vector y, then M is complete, since ℂ is complete. So w exists. Since all elements of M are of the form ay, we can write w = ay for some complex number a. Then (v −ay, y) = 0 or

so

We call a the Fourier coefficient of v with respect to y. Particularly useful is the case where and we can write

26.2.11

Orthogonal Projection

Getting back to the general case, if V = M ⊕ M ⊥ holds, so that to every v there corresponds a unique w ∈ M satisfying (v −w) ∈ M ⊥ the map v → w is called orthogonal projection of V onto M and will be denoted by π .

26.3 26.3.1

The Riesz Representation Theorem for Hilbert Spaces Linear and Antilinear Maps

Let V and W be two complex vector spaces. A map

is called linear if

and is called anti-linear if

26.3.2

Linear Functions

If ℓ: V → C is a linear map (also known as a linear function), then

has codimension one (unless ℓ ≡ 0). Indeed, if

then

and for any z ∈ V,

If V is a normed space and ℓ is continuous, then ker (ℓ) is a closed subspace. The space of continuous linear functions is denoted by V∗. It has its own norm defined by

Suppose that H is a pre-Hilbert space. Then we have an anti-linear map

The Cauchy-Schwarz inequality implies that

and in fact

shows that

In particular, the map ϕ is injective. 26.3.3

The Riesz Representation Theorem

The Riesz representation theorem says that if H is a Hilbert space, then this map is surjective. Theorem26.3.1. Every continuous linear function on H is given by scalar product by some element of H. 26.3.4

Proof of the Riesz Representation Theorem

The proof follows from the theorem about projections applied to

Proof. If ℓ = 0 there is nothing to prove. If ℓ ≠ 0 then N is a closed subspace of codimension one. Choose v ∉ N. Then there is an x ∈ N with (v − x) ⊥ N. Let

Then

For any f ∈ H,

so

or

so if we set

then

for all f ∈ H. The Riesz representation theorem is one of the main heros of this book— despite the fact that its proof is so easy. 26.4 26.4.1

The Riemann-Lebesgue Lemma The Setup

I will consider functions on ℝ which are “integrable” and have the property that for any > 0 there is a step function g such that

This definition depends on the meaning of the word “integrable.” The theory of the Lebesgue integral says that all functions which are Lebesgue integrable will have this property. Clearly, any f which is piecewise continuous and vanishes outside a finite interval has this property. If f is only defined on some interval [a, b] and has this property there, we just extend f by declaring it to be zero outside [a, b] and the extended function still has this property. We will denote our class of functions by L1(ℝ) or simply by L1.

26.4.2

The Averaging Condition

A bounded integrable function h is said to satisfy the averaging condition if

For example, h(t) = sin t satisfies the averaging condition. 26.4.3

The Riemann-Lebesgue Lemma

Theorem 26.4.1 (The Riemann-Lebesgue lemma). If f ∈ L1(ℝ) and h satisfies the averaging condition, then

for any interval [a, b]. Clearly it is enough to prove this for a = 0, b=∞. Proof. If f = 1[c,d], where 0 ≤ c ≤ d < ∞ then

By linearity, the theorem is true for step functions. Choose C such that |h(x)| ≤ C for all x. Let f ∈ L1(ℝ). Choose a step function g such that ℝ | f − g|dx Choose Ωsuch that for all Then for ω > Ω we have

26.5

The Riesz Representation Theorem for Measures

Let X be a locally compact Hausdorff space and L = Cc(X) the space of continuous (complex valued) functions of compact support on X with the uniform norm. For our purposes we may take X to be the space ℝ of real

numbers. The Riesz representation theorem that we need is Theorem 26.5.1. Let F be a bounded linear function on L (with respect to the uniform norm). Then there is a complex valued measure μ on X such that

More precisely, X has a σ-algebraΩcontaining all Borel sets and a unique complex valued measure μ on Ω such that (26.12) holds. Any vector space over the complex numbers can be considered as a vector space over the real numbers, and any (complex) linear function G can be written as G = F1 + i F2 where F1 and F2 are real linear functions. So it is enough to prove the theorem where L is the space of continuous real valued functions of compact support. F is called positive if F (f) ≥ 0 when f ≥ 0.We will prove below that any bounded real valued functional can be written as the difference of two positive functionals. So it is enough to prove the theorem for the case that L consists of real valued functions and F is positive. The proof given here for positive F is essentially taken from [21] and included here for the reader’s convenience. We start with some facts from topology: 26.5.1 Partitions of Unity (in the Topological Sense) and Urysohn’s Lemma We assume the following is known, [21] pp. 39-40: Proposition 26.5.1. Let X be a locally compact Hausdorff space, and let {Uα, α ∈ A} be an open cover of X. Then there exists a partition of unity subordinate to this cover, i.e., a set of continuous functions uα : X ↦ [0, 1] such that supp uα ⊂ Uα and

Suppose that K is a compact subset of X and U is an open subset of X containing K. Consider the two element cover {U, X \ K} and let {f, g} be a partition of unity subordinate to this cover. Since g ≡ 0 on K we deduce from f +

g ≡ 1 that f ≡ 1 on K and f ≤ 1. This gives Urysohn’s lemma: Proposition 26.5.2. If K is a compact subset of X and U an open set containing K there is a continuous function f : X →[0, 1] such that

In fact, the logic of the situation is the reverse: (see [21] as above) in that we first prove Urysohn’s lemma and use it to get a partition of unity. 26.5.2

Urysohn Implies Uniqueness

Suppose we tentatively define

for each open set U ⊂ X. It is not clear, at the moment, how to extend μ to a suitable σ-algebra so as to get a measure and this will require a lot of work. But let us tentatively define

for any subset E ⊂ X. Clearly this coincides with the definition (26.13) on open sets. Lemma 26.5.1. Let U ⊂ X be open with μ(U) < ∞. Then

Proof. Let c < μ(U) so there is an f ∈ L with c < F(f) ≤ 1V. Let K = supp(f) so K ⊂ U. Let W be any open set containing K. So F(f) ≤ μ(W) and by the definition of μ(K) we have F(f) ≤ μ(K). So c ≤ μ(K) ≤ μ(U). We still must define an appropriate σ-algebra extending the definition of μ. But suppose we had a measure ν such that

holds for all compactly supported functions. Let K ⊂ X be compact, and U ⊂ X an open set containing K. By Urysohn there is a function f with 1K ≤ f ≤ 1U so

By the argument of the lemma we conclude that ν(U) = μ(U) (as defined above) so ν = μ on all open sets, hence on all Borel sets. 26.5.3

Tentatively Defining the Appropriate σ -Algebra

Taking our cue from Lemma 26.5.1, let us define, for any subset E ⊂ X,

define ΩB to be the collection of all subsets E such that

and

and define Ω to be the collection of all E ⊂ X such that

We need to prove that Ω is a σ-algebra and this will take some work. But once we prove this, Lemma 26.5.1 will tell us that Ω contains all Borel sets. Proposition 26.5.3. Let K be a compact subset of X and 1k ≤ f, f ∈ L. Then μ(K) ≤ F(f) so K ∈ Ω. Proof. Let 0 < c < 1, let f be as in the proposition, and define

So K ⊂ Vc and cg < f whenever g ∈ L satisfies g ≤ 1Vc. So

Since this holds for all 0 < c < 1, the proposition follows. Proposition 26.5.4. Let K ⊂ X be compact. Then

Proof. The previous proposition tells us that

We must prove the reverse inequality. Here Urysohn comes to the rescue: By definition there is, for any > 0, an open set U ⊂ K with μ(U) ≤ μ(K)+so an f ∈ L with 1K ≤ f ≤ 1U. Then

and this holds for any> 0. Proposition 26.5.5. Let U1, . . . ,UN be a finite collection of open sets and

Then

By induction, it suffices to prove this for N = 2. Proof. Choose g ∈ L, g ≤ 1U1∪U2 . By the partition of unity theorem applied to U1 ∪U2 there are u1, u2 ∈ L such that u1 ≤ 1U1, u2 ≤ 1U2 and u1 +u2 ≡ 1 on U1 ∪U2. So

so

Since this holds for all such g, the proposition follows for N = 2 hence for all N. 26.5.4

Countable Sub-additivity of μ in General

Proposition 26.5.6. Let Ei be a countable collection of subsets of the X and

Then

If one of the μ(Ei)=∞ there is nothing to prove. So we must prove this in the case that all the μ(Ei) < ∞. For this we apply the trick: Proof. Choose Un ⊃ En such that μ(Un) < ∞ and

Let U ≔ Un and f ∈ L, f ≤ 1U. So supp f ⊂ U, i.e., is covered by the Ui. Since supp f is compact, it is covered by a finite collection of the Ui so

for some N. So

for any ∈ > 0. 26.5.5

Countable Additivity for Elements of B

Proposition 26.5.7. Let E1, E2, . . . be pairwise disjoint members of ΩB, and E ≔ Ei. Then

If μ(E) < ∞ then E ∈ ΩB. If μ(E)=∞ then this follows from Proposition 26.5.6. So we may assume that μ(E) < ∞. Again we apply the trick: Proof. For > 0 choose Kn ⊂ En such that μ(Kn) ≥ μ(En)−2−n∈. We can do so by the definition of ΩB. Let MN ≔ K1 ∪· ··∪ KN. Then

Since this holds for all N we conclude that Since this holds for all> 0 we conclude that Combined the general countable subaddivity this gives the equality in the proposition. So we can find N such that so

and (26.15) holds. Let E ∈ ΩB so there exists an open set U ⊃ E and a compact set K ⊂ E such that

By Lemma 26.5.1 V \ K ∈ ΩB and by additivity

so we get the “Lebesgue condition”: Proposition 26.5.8. Let E ∈ ΩB. For any> 0 there exist compact subsets K ⊂ E and open sets U ⊃ E such that

26.5.6

Proof That ΩB Is a σ -algebra

Proposition 26.5.9. Let A, B ∈ ΩB. Then A\ B, A ∪ B and A ∩ B belong to ΩB. Proof. We start with A \ B:Choose K ⊂ A ⊂ U, K'⊂ B ⊂ U' as in Proposition 26.5.8. We have

so μ(A \ B) ≤ μ(K \U')+2and K \U' is compact so A \ B ∈ ΩB.

Then A ∪ B = (A \ B) ∪ B and these are disjoint sets belonging to ΩB so A ∪ B ∈ ΩB. Similarly, A ∩ B = A \ (A \ B). As a consequence, we will prove that: Proposition 26.5.10. Ω is a σ-algebra containing the Borel sets. Clearly, Ω contains all compact sets hence all Borel sets. Let A ∈ Ω and K ⊂ X be compact. Then (X \ A) ∩ K = K \ A so (X \ A) ∩ K ∈ ΩB, so by the preceding proposition X \ A ∈ Ω. Suppose that

Let B1 ≔ A1 ∩ K and

The Bn are pairwise disjoint and their union is A ∩ K. So A ∩ K ∈ ΩB and by the definition of Ω this says that A ∈ Ω. So Ω is a σ-algebra. If C ⊂ X is closed, and K ⊂ X is compact, then K ∩C is compact, hence belongs to ΩB, hence C ∈ Ω. Thus Ω contains all closed subsets hence contains all Borel sets. Lemma 26.5.2. ΩB = {E ⊂ Ω | μ(E) < ∞}. Proof. If E ∈ ΩB then so is E ∩ K for any compact set K so E ∈ Ω and by definition of ΩB, μ(E) < ∞. Conversely, suppose E ∈ Ω and μ(E) < ∞. For> 0 choose an open U ⊃ E and K ⊂ E open with μ(U) < ∞and μ(U \ K) < . Now E ∩ K ∈ ΩB so we can find a compact H ⊂ X with H ⊂ E ∩ K and μ(E ∩ K) ≤ μ(H)+. SinceE ⊂ (E ∩ K) ∪ H we get

Theorem 26.5.2. μ is a measure on Ω. It is the unique measure with

We already know from Proposition 26.5.7 and Lemma 26.5.2 that μ is a measure on Ω. We must check (26.16). We approximate, as usual, by step functions. Let f ∈ L and K a compact set containing supp f. Let [a, b] be an interval containing f (K). Let

be a partition of [a, b]. The sets En ≔ f −1((yn, yn+1]) are pairwise disjoint with union K. There are open sets Un ⊃ En with

Let {un} be a partition of unity subordinate to the cover {Un} of K. So and

Now un f ≤ (yn +)un and yn −≤ f (x) ∀x ∈ En. So

so

So

But the definition of the integral associated with a measure as the limit of the sum of the integral of step functions ≤ f shows that

Thus

Applied to − f gives

Together these imply (26.16). This concludes the proof of the Riesz representation theorem for positive F. We now need a decomposition theorem to conclude the Riesz representation theorem in general. 26.5.7

A Decomposition Theorem

Let F be a bounded linear function on L (relative to the sup norm). If f ≥ 0 define

Since g = 0 is a possible candidate,

Also F(g) ≤ |F(g)| ≤ ∥F∥∥g∥ ≤ ∥F∥∥ f ∥ for all 0 ≤ g ≤ f where we are taking sup norms. So

Clearly F+(cf) = ncF+(f) for all c ≥ 0. Suppose that f1 and f2 are both non-negative elements of L, and 0 ≤ g1 ≤ f1 and 0 ≤ g2 ≤ f2. Then

On the other hand, if 0 ≤ g ≤ (f1 + f2) then 0 ≤ f1 ∧ g and g ∧ f1 ∈ L and vanishes at points x where g(x) ≤ f1(x) while at points where g(x) > f1(x) we have g(x) = g − g(x)∧ f1(x) = g(x)− f1(x) ≤ f2(x). So

So if f1 and f2 are non-negative.

Now write any f ∈ L as f1 − g1 where f1 and g1 are non-negative. For example, we could take f1 = f + and g1 = f−. Define

This is well defined, for if we also had f = f2 − g2 then

so

so

From this it follows that F + so extended is linear, and

so F + so extended is a bounded linear function which is non-negative. Define F− ≔ F + − F. It is a bounded linear function. By its definition, F+(f) ≥ F(f) if f ≥ 0. So F−(f) ≥ 0 if f ≥ 0. We have proved: Proposition 26.5.11. Every bounded linear function F on L can be written as F = F + − F− where F + and F− are bounded positive linear functions. We conclude with the desired: Theorem 26.5.3. Any bounded (real valued) linear function F can be written as

where μ+ and μ− are non-negative Borel measures.

Bibliography [1] Agmon, Shmuel Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations Princeton University Press (1983) [2] Chernoff, Paul “Note on Product Formulas for Operator Semigroups,” J. Functional Analysis 2 (1968) 238–242 [3] Coulson, Charles Alfred Coulson’s Valence Oxford Univ. Press (1980) [4] Davies, E.B. Spectral Theory and Differential Operators Cambridge University Press (1995) [5] Fréchet, Maurice “Sur les ensembles de fonctions et les operations lineares” Comptes Rendus 144 (1907) 1414–1416 [6] Helffer, Bernard Spectral Theory and Its Applications Cambridge University Press (2013) [7] Hille, Einer Functional Analysis and Semi-Groups Am. Math. Soc. Colloq. Pub. 31 (1948) [8] Hislop, P.D. and Segal, I.M. Introduction to Spectral Theory Springer (1996) [9] Ikebe, Teruo “Eigenfunction Expansions Associated with the Schrödinger Operators and Their Applications to Scattering Theory,” Arch. Rational Mech. Anal. 5 (1960) 1–34 [10] Kantorovitz, Shmuel Topics in Operator Semigroups Springer (2009) [11] Lax, Peter and Phillips, Ralph Scattering Theory Academic Press (1990) [12] Lions, G. and Vergne, M. The Weil Representation, Maslov Index and Theta Series Springer (1980) [13] Loomis, Lynn An Introduction to Abstract Harmonic Analysis van Nostrand (1953) [14] Lorch, Edgar Raymond Spectral Theory Oxford University Press (1962) [15] Lunardi, Allessandra Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications Birkhauser (1995) Second edition: 2013 reprint of the 1995 original [16] Mackey, GeorgeW. Mathematical Foundations of Quantum Mechanics Dover (2004) [17] Reed, Michael and Simon, Barry Methods of Mathematical Physics 4 volumes, Academic Press (1980) [18] Riesz, Frederic “Sur une espece de geometrie analytique des systemes de functions summables,” Comptes Rendus 144 (1907) 1409–1411 [19] Riesz, Frederic “Sur les operations functionelles lineares,” Comptes Rendus 149 (1909) 974–977 [20] Roach, Gary An Introduction to Echo Analysis, Scattering Theory and Wave Propagation Springer (2008) [21] Rudin, Walter Real and Complex Analysis 3rd ed., McGraw-Hill (1987) [22] Berezin and Shubin The Schrödinger Equation Kluger (1981) [23] Singer, Stephanie Linearity, Symmetry, and Prediction in the Hydrogen Atom Springer (2005) [24] Stone, Marshal Linear Transformations in Hilbert Space and Their Applications to Analysis American Mathematical Society (1932)

[25] Teschl, Gerard Mathematical Methods in Quantum Mechanics American Mathematical Society, Vol. 157 (2009, 2014) [26] Weyl, Hermann “Uber beschrankte quadratische Formen, deren Differenz vollstetig ist” Rend. Circ. Mat. Palermo 27 (1909) pp. 373–392 [27] Wintner, Aurel Spektraltheorie der Unendlichen Matrizen S. Hirse Verlag (1929) [28] Yoshida, K. Functional Analysis Springer (1995)

Index A-bounded operator, 92 A-compact, 101 Agmon metric, 210 Agmon’s theorem, 211 almost holomorphic extensions, 82 Amrein-Georgescu theorem, 203 approximate eigenvalue, 94, 95, 97 Bohr radius, 157 Borel transform, 132 Born approximation, 97, 261 Born series, 261 Born-Oppenheimer approximation, 157 Breit-Wigner function, 237 Cauchy principal value, 16 Chernoff, 265, 276 closed linear transformations, 32 convolution, 6 core, 92 covalent bond, 159 cyclic vector, 141 d’Alembert solution, 246 Davies’ proof, 85 density of states, 118 Dirichlet integral, 17 dissociation energy, 158 domain of a graph, 32 Duhamel’s principle, 248 Dynkin-Helffer-Sjöstrand formula, 79, 83 essential spectrum, 98 exponential series, 19 Feynman-Kac formula, 284

finite rank operators, 109 first resolvent identity, 40, 47, 73, 96 Fourier inversion, 9 Fourier transform, 5 fractional power, 145 Fréchet, 56 Fréchet space, 5, 64 Friedrichs extension, 149 functional calculus, 20, 77 Gaussian, 7 Gelfand’s formula, 222 Gelfand’s proof, 22 Gelfand’s theorem, 26 Glazman’s lemma, 173 graph of a linear transformation, 32 Groenewold-van Hove theorem, 265 Hardy’s inequality, 150 Heisenberg Uncertainty Principle, 12 Heitler-London theory, 158 Helly selection principle, 134 Herglotz functions, 132 Hilbert Schmidt operators, 112, 114 Hille-Yosida theorem, 71–73 Huckel theory, 163 Huygens’ principle, 251 incoming and outgoing spaces, 232 infinitesimal generator, 64 Jost solutions, 182 Kato potentials, 205 Kato-Rellich theorem, 93, 196 Krein-Rutman theorem, 171 Laplace transform, 15 Lax-Milgram theorem, 146–148 Lorch’s proof, 217 Lumer-Phillips theorem, 89 map of a graph, 32 mean ergodic theorem, 194, 196, 203 Mellin inversion formula, 18, 72 modified contour, 42

multiplication formula, 8 multiplication operators, 58 no crossing rule, 163 non-negative operators, 28, 144 normal operators, 23 Nyquist rate, 12 partitions of unity, 137, 298 Plancherel’s theorem, 9 Poisson algebra, 265 Poisson summation formula, 10 positivity improving operators, 171 positivity preserving operators, 171 putative resolvent, 64 quantum tunneling, 210 Radon transform, 250 RAGE theorem, 194, 198 Rayleigh-Ritz, 154, 170 rectangular barrier, 180 relative bound, 93 relatively compact, 116 resolutions of the identity, 137 resolvent, 24, 33 resolvent set, 24, 33 Riesz-Dunford calculus, 217 Riesz representation theorem for Hilbert space, 56, 96, 128, 129, 136, 146, 147, 295 Riesz representation theorem for measures, 128, 298 Ruelle’s theorem, 193, 208 scattering matrix, 186 scattering operator, 255 second resolvent identity, 40 sectorial operator, 42 secular equation, 161 self-adjoint operator, 57 Shannon sampling theorem, 10 Sinai representation theorem, 240 singular values, 111 singular Weyl sequence, 115, 116 spectral radius, 25 spectrum, 33

spreading of the wave packet, 176 square well, 91, 104, 105 stability of matter, 125 stationary Schrödinger equation, 179 Stieltjes inversion formula, 133, 135 Stieltjes transform, 132 Stone’s formula, 139, 223 Stone’s theorem, 55, 69, 76 Stone-von Neumann theorem, 241 Sturm comparison theorem, 165 Sturm oscillation theorem, 165 symmetric operator, 57 tempered distributions, 13 Trotter product formula, 279, 282 Urysohn’s lemma, 299 valence, 162 virial theorem, 124 wave operators, 253 Weyl’s law, 118, 122 Weyl’s theorem, 91, 101 Wronskians, 183 Yukawa potential, 188

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