On the Feynman-Kac formula for Schrödinger semigroups on vector bundles [Elektronische Ressource] / Batu Güneysu

On the Feynman-Kac formula forSchr¨odinger semigroups on vectorbundlesDissertationzurErlangung des Doktorgrades (Dr. rer. nat)derMathematisch-Naturwissenschaftlichen Fakult¨atderRheinischen Friedrich-Wilhelms-Universit¨at Bonnvorgelegt vonBatu Gune¨ ysuaus AschaffenburgBonn, Dezember 2010AngefertigmitGenehmigungderMathematisch-NaturwissenschaftlichenFakult¨atder Rheinischen Friedrich-Wilhelms-Universit¨at Bonn1. Gutachter: Prof. Dr. Matthias Lesch2. Gutachter: Prof. Dr. Andreas EberleTag der Promotion: 15.04.2011Erscheinungsjahr: 2011AbstractIn this thesis we generalize the Feynman-Kac formula to semigroups thatcorrespond to Schr¨odinger type operators with possibly singular potentialson vector bundles over noncompact Riemannian manifolds.Thisprobabilisticformulaisthenusedtoobtaininformationaboutthespec-tral theory of these operators.A first class of applications corresponds to semigroup domination: We showhow the spectrum can be estimated by usual scalar Schr¨odinger operatorson functions. This includes estimates for the bottom of the spectrum and,from a Brownian bridge version of our Feynman-Kac formula, we also obtainestimates for the integral kernel and the trace of the semigroup.