## Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration ApproachA 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. |

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### Contents

Basic Assumptions of Stochastic Spectral Analysis Free Feller Operators | 1 |

B Assumptions and Free Feller Generators | 5 |

C Examples | 11 |

D Heat kernels | 25 |

E Summary of Schrödinger semigroup theory | 33 |

E2 Brownian motion and related processes | 39 |

E3 KatoFeller potentials for the Laplace operator | 47 |

E4 Schrödinger semigroups | 49 |

C Singular Perturbations | 217 |

Convergence of Resolvent Differences | 233 |

Spectral Properties of Selfadjoint Feller Operators | 257 |

A Qualitative spectral results | 259 |

B Quantitative estimates for regular potentials | 281 |

C Quantitative estimates for singular potentials in terms of the weighted Laplace transform of the occupation time for large coupling parameters | 309 |

C1 Estimates for the Laplace transform of the occupation time for Wiener processes | 314 |

C2 Quantitative large coupling estimates for Feller operators in terms of the weighted Laplace transform of the occupation time | 318 |

E5 Generalizations and modifications | 51 |

Perturbations of Free Feller Operators | 53 |

Framework of stochastic spectral analysis | 56 |

A Regular Perturbations | 57 |

B Integral kernels martingales pinned measures | 76 |

C Singular perturbations | 87 |

Proof of Continuity and Symmetry of FeynmanKac Kernels | 103 |

Resolvent and Semigroup Differences for Feller Operators Operator Norms | 129 |

B Singular perturbations | 145 |

HilbertSchmidt Properties of Resolvent and Semigroup Differences | 161 |

B Singular perturbations | 182 |

Trace Class Properties of Semigroup Differences | 205 |

B Regular Perturbations | 210 |

Spectral Theory | 328 |

Semigroup Theory | 340 |

Markov Processes Martingales and Stopping Times | 348 |

Dirichlet Kernels Harmonic Measures Capacities | 368 |

D1 Continuity and symmetry of Dirichlet kernels | 369 |

D2 Harmonic measures and equilibrium potentials | 388 |

D3 Capacities | 400 |

Dinis Lemma Scheffes Theorem Monotone Class Theorem | 418 |

References | 422 |

436 | |

446 | |

### Other editions - View all

Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional ... Michael Demuth,Jan A. van Casteren No preview available - 2000 |

Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional ... Michael Demuth,Jan A. van Casteren No preview available - 2012 |