On generalized Schrödinger semigroups

2011/09/01 by Güneysu, Batu
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.1109.0151

Abstract

We extend the Feynman-Kac formula for Schrödinger 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 L² -> L^p 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. These results in particular apply to a very general class of magnetic Schrödinger operators on Riemannian manifolds.

Related

• Covariant Schrödinger semigroups on Riemannian manifolds
• A Feynman-Kac Formula for Unbounded Semigroups
• Continuity properties of Schrödinger semigroups with magnetic fields
• The Schrödinger semigroup on some flat and non flat manifolds
• Path integrals and the essential self-adjointness of differential operators on noncompact manifolds