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9783764358877

Stochastic Spectral Theory for Selfadjoint Feller Operators

by ;
  • ISBN13:

    9783764358877

  • ISBN10:

    3764358874

  • Format: Hardcover
  • Copyright: 2000-09-01
  • Publisher: Birkhauser

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Summary

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.

Table of Contents

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

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