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Compactness of Schrödinger semigroups
Author(s) -
Lenz Daniel,
Stollmann Peter,
Wingert Daniel
Publication year - 2010
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200910054
Subject(s) - compact space , mathematics , semigroup , bounded function , measure (data warehouse) , pure mathematics , context (archaeology) , dirichlet distribution , discrete mathematics , mathematical analysis , boundary value problem , computer science , paleontology , database , biology
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)