On the Feynman-Kac formula for Schrödinger semigroups on vector bundles

Abstract

In this thesis we generalize the Feynman-Kac formula to semigroups that correspond to Schrödinger type operators with possibly singular potentials on vector bundles over noncompact Riemannian manifolds. <br /> This probabilistic formula is then used to obtain information about the spectral theory of these operators. <br /> A first class of applications corresponds to semigroup domination: We show how the spectrum can be estimated by usual scalar Schrödinger operators on functions. This includes estimates for the bottom of the spectrum and, from a Brownian bridge version of our Feynman-Kac formula, we also obtain estimates for the integral kernel and the trace of the semigroup. <br /> As another application of the Feynman-Kac formula, we introduce the class of Kato potentials on vector bundles and use probabilistic methods to prove that the semigroups corresponding to Schrödinger type operators with local Kato potentials map square integrable sections to bounded continuous sections. In particular, this implies the boundedness and the continuity of the eigensections of these operators. <br /> We finally specify some of these results to Schrödinger type operators on trivial vector bundles.