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Compactness of Schrödinger semigroups

✍ Scribed by Daniel Lenz; Peter Stollmann; Daniel Wingert

Publisher
John Wiley and Sons
Year
2010
Tongue
English
Weight
150 KB
Volume
283
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of self‐adjoint operators that are bounded below (on an L^2^‐space).

For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of the semigroup in terms of relative compactness of the operators of multiplication with characteristic functions of sublevel sets. In the context of Dirichlet forms, we can even characterize compactness of the semigroup for measure perturbations. Here, certain ‘averages’ of the measure outside of compact sets play a role.

As an application we obtain compactness of semigroups for Schrödinger operators with potentials whose sublevel sets are thin at infinity (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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