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The spectral shift function for compactly supported perturbations of Schrödinger operators on large bounded domains

Authors: Peter D. Hislop and Peter Müller
Journal: Proc. Amer. Math. Soc. 138 (2010), 2141-2150
MSC (2010): Primary 81U05, 35P15, 47A40; Secondary 47A75
Published electronically: February 9, 2010
MathSciNet review: 2596053
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Abstract: We study the asymptotic behavior as $ L \rightarrow \infty$ of the finite-volume spectral shift function for a positive, compactly supported perturbation of a Schrödinger operator in $ d$-dimensional Euclidean space, restricted to a cube of side length $ L$ with Dirichlet boundary conditions. The size of the support of the perturbation is fixed and independent of $ L$. We prove that the Cesàro mean of finite-volume spectral shift functions remains pointwise bounded along certain sequences $ L_n \rightarrow \infty$ for Lebesgue-almost every energy. In deriving this result, we give a short proof of the vague convergence of the finite-volume spectral shift functions to the infinite-volume spectral shift function as $ L \rightarrow \infty$. Our findings complement earlier results of W. Kirsch [Proc. Amer. Math. Soc. 101, 509-512 (1987); Int. Eqns. Op. Th. 12, 383-391 (1989)], who gave examples of positive, compactly supported perturbations of finite-volume Dirichlet Laplacians for which the pointwise limit of the spectral shift function does not exist for any given positive energy. Our methods also provide a new proof of the Birman-Solomyak formula for the spectral shift function that may be used to express the measure given by the infinite-volume spectral shift function directly in terms of the potential.

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Additional Information

Peter D. Hislop
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Peter Müller
Affiliation: Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstraße 39, 80333 München, Germany

Received by editor(s): September 3, 2009
Published electronically: February 9, 2010
Additional Notes: The first author was supported in part by NSF grant 0503784 while this work was being done.
Communicated by: Walter Craig
Article copyright: © Copyright 2010 American Mathematical Society

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