  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.
