Continuous integral kernels for unbounded Schroedinger semigroups and their spectral projections

TL;DR

This paper extends a Feynman-Kac formula to analyze unbounded Schrödinger semigroups with magnetic potentials, establishing their integral kernels and spectral projections as continuous Carleman operators, with applications to disordered systems.

Contribution

It introduces a generalized Feynman-Kac framework for unbounded Schrödinger operators, characterizes their semigroups and spectral projections as continuous kernels, and applies these results to random potentials.

Findings

Semigroups generated by unbounded Schrödinger operators have continuous integral kernels.

Spectral projections of these operators are also Carleman operators with continuous kernels.

The results justify integral-kernel representations of the density of states in disordered systems.

Abstract

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of the generating Schroedinger operator also act as Carleman operators with continuous integral kernels. Applications to Schroedinger operators with rather general random scalar potentials include a rigorous justification of an integral-kernel representation of their integrated density of states - a relation frequently used in the physics literature on disordered solids.