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.