A Mathematical Companion to Quantum Mechanics 1st Edition by Shlomo Sternberg ISBN 0486839826 9780486839820

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ISBN 10: 0486839826
ISBN 13: 9780486839820
Author: Shlomo Sternberg

This original 2019 work, based on the author’s many years of teaching at Harvard University, examines mathematical methods of value and importance to advanced undergraduates and graduate students studying quantum mechanics. Its intended audience is students of mathematics at the senor university level and beginning graduate students in mathematics and physics. Early chapters address such topics as the Fourier transform, the spectral theorem for bounded self-joint operators, and unbounded operators and semigroups. Subsequent topics include a discussion of Weyl’s theorem on the essential spectrum and some of its applications, the Rayleigh-Ritz method, one-dimensional quantum mechanics, Ruelle’s theorem, scattering theory, Huygens’ principle, and many other subjects.

A Mathematical Companion to Quantum Mechanics 1st 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 Self-adjoint 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

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

13.1.1 Cyclic Vectors

13.1.2 The Isometry U in the Cyclic Case

13.1.3 The General Case

13.2 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 Non-negative Self-adjoint Operator

13.3 The Lax-Milgram Theorems

13.3.1 Gelfand’s Rigged Hilbert Spaces

13.3.2 The Hermitian Case—Lax-Milgram-III

13.4 Semi-bounded Operators and the So-called Friedrichs Extension

13.4.1 Semi-bounded Operators

13.4.2 The Friedrichs Extension

13.5 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

15.2.3 Comparison of Two Equations

15.2.4 Application to Eigenvectors

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

15.3.1 The Quadratic Form H(f, g)

15.4 Perron-Frobenius-Krein-Rutman

15.5 Variational Principles, Glazman’s Lemma

15.6 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

19.3 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 Sub-algebra 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

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