ON SPECTRAL THEORY OF SELFADJOINT FELLER GENERATORS

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Abstract
Upon introducing the theory of (symmetric) Feller semigroups, regularly and singularly perturbations of generators of Feller semigroups are studied. For the regular perturbations potentials belonging to the Kato-Feller class are admitted. The singular perturbations arise from potential barriers with increasing height. The Feynman-Kac representation is derived for the singular case. Selfadjointness conditions are given for these perturbed generators of Feller semigroups. Up to some extent we give explicit forms. The second main part consists of spectral theoretical considerations for these perturbed Feller generators. We give some examples which indicate the usefulness of the Feynman-Kac formula in spectral theory. We are interested in compactness properties of semigroup differences, which consist of trace class or Hilbert-Schmidt operators.

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