On generalized Schr\"odinger semigroups

TL;DR

This paper extends the Feynman-Kac formula to handle very singular potentials in Schr"odinger operators on vector bundles over noncompact Riemannian manifolds, enabling new L^p and smoothing results for these semigroups.

Contribution

It introduces a generalized path integral formula for Schr"odinger operators with singular potentials, broadening the scope of analysis on Riemannian manifolds.

Findings

Established L^p bounds on ground state energies.

Proved L^2 to L^p smoothing properties of Schr"odinger semigroups.

Demonstrated L^2 boundedness and smoothing for magnetic Schr"odinger operators.

Abstract

We extend the Feynman-Kac formula for Schr\"odinger type operators on vector bundles over noncompact Riemannian manifolds to possibly very singular potentials that appear in hydrogen like quantum mechanical problems and that need not be bounded from below or locally square integrable. This path integral formula is then used to prove several L^p-type results, like bounds on the ground state energy and L^2 -> L^p smoothing properties of the corresponding Schr\"odinger semigroups. As another main result, we will prove that with a little control on the Riemannian structure, the latter semigroups are also L^2->{bounded continuous} smoothing for Kato decomposable potentials. These results in particular apply to a very general class of magnetic Schr\"odinger operators on Riemannian manifolds.