04-68 Kurt Broderix, Hajo Leschke, Peter Mueller

Continuous integral kernels for unbounded Schroedinger semigroups and their spectral projections (325K, Postscript) Mar 8, 04

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Abstract. By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc. \textbf{28} (2000), 317--321], we study one-parameter semigroups generated by (the negative of) rather general Schr{\"o}dinger 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 Schr\"odinger operator also act as Carleman operators with continuous integral kernels. Applications to Schr{\"o}dinger 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.

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