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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.
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.
The book provides a systemic treatment of time-dependent strong
Markov processes with values in a Polish space. It describes its
generators and the link with stochastic differential equations in
infinite dimensions. In a unifying way, where the square gradient
operator is employed, new results for backward stochastic
differential equations and long-time behavior are discussed in
depth. The book also establishes a link between propagators or
evolution families with the Feller property and time-inhomogeneous
Markov processes. This mathematical material finds its applications
in several branches of the scientific world, among which are
mathematical physics, hedging models in financial mathematics, and
population models.
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