Continuous integral kernels for unbounded Schrödinger semigroups and their spectral projections

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc. 28, 317–321 (2000)], we study one-parameter semigroups generated by (the negative of) rather general Schrödinger 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 Schrödinger operator also act as Carleman operators with continuous integral kernels. Applications to Schrödinger 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 systems.