Stochastic Spectral Theory for Selfadjoint

A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. For such operators regular and singular perturbations of order zero and their spectral properties are investigated.A complete treatment of the Feynman-Kac formula is given. The theory is applied to such topics as compactness or trace class properties of differences of Feynman-Kac semigroups, preservation of absolutely continuous and/or essential spectra and completeness of scattering systems.The unified approach provides a new viewpoint of and a deeper insight into the subject. The book is aimed at advanced students and researchers in mathematical physics and mathematics with an interest in quantum physics, scattering theory, heat equation, operator theory, probability theory and spectral theory.

Preface

vii

A readers guideline

x

Acknowledgement

xi

Basic Assumptions of Stochastic Spectral Analysis: Free Feller Operators

1

(52)

Introduction

1

(4)

Assumptions and Free Feller Generators

5

(6)

Examples

11

(14)

Heat Kernels

25

(8)

Summary of Schrodinger semigroup theory

33

(20)

Gaussian processes

33

(6)

Brownian motion and related processes

39

(8)

Kato-Feller potentials for the Laplace operator

47

(2)

Schrodinger semigroups

49

(2)

Generalizations and modifications

51

(2)

Perturbations of Free Feller Operators

53

(50)

The framework of stochastic spectral analysis

56

(1)

Regular perturbations

57

(19)

Integral kernels, martingales, pinned measures

76

(11)

Singular perturbations

87

(16)

Proof of Continuity and Symmetry of Feynman-Kac Kernels

103

(26)

Resolvent and Semigroup Differences for Feller Operators: Operator Norms

129

(32)

Regular perturbations

129

(16)

Singular perturbations

145

(16)

Hilbert-Schmidt Properties of Resolvent and Semigroup Differences

161

(44)

Regular perturbations

161

(21)

Singular perturbations

182

(23)

Trace Class Properties of Semigroup Differences

205

(28)

General trace class criteria

205

(5)

Regular perturbations

210

(7)

Singular perturbations

217

(16)

Convergence of Resolvent Differences

233

(24)

Spectral Properties of Self-adjoint Feller Operators

257

(76)

Qualitative spectral results

259

(22)

Quantitative estimates for regular potentials

281

(28)

Quantitative estimates for singular potentials in terms of the weighted Laplace transform of the occupation time (for large coupling parameters)

309

(24)

Estimates for the Laplace transform of the occupation time for Wiener processes

314

(4)

Quantitative large coupling estimates for Feller operators in terms of the weighted Laplace transform of the occupation time

318

(15)

Appendix A Spectral Theory

333

(12)

Appendix B Semigroup Theory

345

(8)

Appendix C Markov Processes, Martingales and Stopping Times

353

(20)

Appendix D Dirichlet Kernels, Harmonic Measures, Capacities

373

(50)

D.1 Continuity and symmetry of Dirichlet kernels

374

(19)

D.2 Harmonic measures and equilibrium potentials

393

(12)

D.3 Capacities

405

(18)

Appendix E Dini's Lemma Scheffe's Theorem Monotone Class Theorem

423

(4)

References

427

(14)

Index of Symbols

441

(10)

Subject Index

451

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Stochastic Spectral Theory for Selfadjoint Feller Operators

Summary

Table of Contents

Supplemental Materials