On generalized Schrödinger semigroups

We prove a Feynman–Kac formula for Schrödinger type operators on vector bundles over arbitrary Riemannian manifolds, where the potentials are allowed to have strong singularities, like those that typically appear in atomic quantum mechanical problems. This path integral formula is then used to prove several Lp-type results, like bounds on the ground state energy and L2⇝Lp smoothing properties of the corresponding Schrödinger semigroups. As another main result, we will prove that with a little control on the Riemannian structure, the latter semigroups are also L2⇝{bounded continuous} smoothing for Kato decomposable potentials.