This textbook provides a graduate-level introduction to the spectral theory of linear operators on Banach and Hilbert spaces, guiding readers through key components of spectral theory and its applications in quantum physics. Based on their extensive teaching experience, the authors present topics in a progressive manner so that each chapter builds on the ones preceding. Researchers and students alike will also appreciate the exploration of more advanced applications and research perspectives presented near the end of the book.
Beginning with a brief introduction to the relationship between spectral theory and quantum physics, the authors go on to explore unbounded operators, analyzing closed, adjoint, and self-adjoint operators. Next, the spectrum of a closed operator is defined and the fundamental properties of Fredholm operators are introduced. The authors then develop the Grushin method to execute the spectral analysis of compact operators. The chapters that follow are devoted to examining Hille-Yoshida and Stone theorems, the spectral analysis of self-adjoint operators, and trace-class and Hilbert-Schmidt operators. The final chapter opens the discussion to several selected applications. Throughout this textbook, detailed proofs are given, and the statements are illustrated by a number of well-chosen examples. At the end, an appendix about foundational functional analysis theorems is provided to help the uninitiated reader.
A Guide to Spectral Theory: Applications and Exercises is intended for graduate students taking an introductory course in spectral theory or operator theory. A background in linear functional analysis and partial differential equations is assumed; basic knowledge of bounded linear operators is useful but not required. PhD students and researchers will also find this volume to be of interest, particularly the research directions provided in later chapters.
Author(s): Christophe Cheverry, Nicolas Raymond
Series: Birkhäuser Advanced Texts Basler Lehrbücher
Edition: 1
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 258
Tags: Spectral Theory, Banach, Hilbert, Operator, Compact, Unbounded, Self-adjoint, Fredholm
FOREWORD
PROLEGOMENA
CONTENTS
Chapter 1. A first look at spectral theory
1.1. From quantum physics to spectral considerations
Examples to keep in mind
A list of questions
A more elaborate statement
1.2. A paradigmatic spectral problem
1.2.1. A question
1.2.2. An answer
1.3. Some density results
Chapter 2. Unbounded operators
2.1. Definitions
2.2. Adjoint and closedness
2.2.1. About duality and orthogonality
2.2.2. Adjoint of bounded operators in Hilbert spaces
2.2.3. Adjoint of unbounded operators in Hilbert spaces
2.2.4. Creation and annihilation operators
2.3. Self-adjoint operators and essentially self-adjoint operators
2.3.1. Symmetric and self-adjoint operators
2.3.2. Essentially self-adjoint operators
2.3.3. Essential self-adjointness of Schrödinger operators
2.4. Polar decomposition
2.5. Lax–Milgram theorems
2.6. Examples
2.6.1. Dirichlet Laplacian
2.6.2. Neumann Laplacian
2.6.3. Harmonic oscillator
2.6.4. Exercise on the magnetic Dirichlet Laplacian
2.7. Regularity theorem for the Dirichlet Laplacian
2.7.1. Difference quotients
2.7.2. Partition of the unity
2.7.3. Local charts
2.7.4. Proof
2.8. Notes
Chapter 3. Spectrum
3.1. Definitions and basic properties
3.1.1. Holomorphic functions valued in a Banach space
3.1.2. Basic definitions and properties
3.1.3. About the bounded case
3.1.4. Spectrum of the adjoint
3.2. Spectral radius and resolvent bound in the self-adjoint case
3.3. Riesz projections
3.3.1. Properties
3.3.2. About the finite algebraic multiplicity
3.4. Fredholm operators
3.4.1. Definition and first properties
3.4.2. Spectrum and Fredholm operators
3.5. Notes
Chapter 4. Compact operators
4.1. Definition and fundamental properties
4.2. Compactness in Lp spaces
4.2.1. About the Ascoli theorem in Lp spaces
4.2.2. Rellich theorems
4.3. Operators with compact resolvent
4.4. Notes
Chapter 5. Fredholm theory
5.1. Grushin formalism
5.2. On the index of Fredholm operators
5.3. On the spectrum of compact operators
5.4. Study of the complex Airy operator
5.5. An application of the Grushin formalism
5.6. Toeplitz operators on the circle
5.7. Notes
Chapter 6. Spectrum of self-adjoint operators
6.1. Compact normal operators
6.2. About the harmonic oscillator
6.2.1. Domain considerations
6.2.2. Spectrum of the harmonic oscillator
6.3. Characterization of the spectra
6.3.1. Properties
6.3.2. Determining the essential spectrum: an example
6.4. Min-max principle
6.4.1. Statement and proof
6.4.2. Sturm–Liouville's oscillation theorem
6.4.3. Weyl's law in one dimension
6.4.4. Proof of the Weyl law
6.4.5. Some exercises
6.5. On the ground-state energy of the hydrogen atom
6.6. Notes
Chapter 7. Hille-Yosida and Stone's theorems
7.1. Semigroups
7.2. Hille–Yosida's theorem
7.2.1. Necessary condition
7.2.2. Sufficient condition
7.3. Stone's theorem
7.3.1. Necessary condition
7.3.2. Sufficient condition
7.4. Notes
Chapter 8. About the spectral measure
8.1. A functional calculus based on the Fourier transform
8.2. Where the spectral measure comes into play
8.2.1. Extending a map
8.2.2. Riesz theorem and spectral measure
8.3. Spectral projections
8.3.1. Properties
8.3.2. Extension to unbounded functions
8.3.3. Characterization of the spectra
8.3.4. Positive and negative parts of a self-adjoint operator
8.3.5. Decomposition of the spectral measure
8.4. Notes
Chapter 9. Trace-class and Hilbert-Schmidt operators
9.1. Trace-class operators
9.2. Hilbert–Schmidt operators
9.2.1. Definition and first properties
9.2.2. Trace of a trace-class operator
9.3. A fundamental example
9.4. Local traces of the Laplacian
9.4.1. The case of mathbbRd
9.4.2. The case of mathbbRd+
9.5. Notes
Chapter 10. Selected applications of the functional calculus
10.1. Lieb's Variational Principle
10.1.1. Statement
10.1.2. Illustration
10.2. Stone's formula
10.2.1. Statement
10.2.2. A criterion for absolute continuity
10.3. Elementary Mourre's theory and Limiting Absorption Principle
10.3.1. Mourre estimates
10.3.2. Limiting Absorption Principle and consequence
10.3.3. Example of Mourre estimates
10.4. Notes
Chapter A. Reminders Of Functional Analysis
A.1. Hahn–Banach theorem
A.2. Baire theorem and its consequences
A.3. Ascoli Theorem
A.4. Sobolev spaces
A.5. Notes
Bibliography
Index
