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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Small perturbations and the eigenvalues of the Laplacian on large bounded domains

Author: Werner Kirsch
Journal: Proc. Amer. Math. Soc. 101 (1987), 509-512
MSC: Primary 35P20; Secondary 35J25, 47F05
MathSciNet review: 908658
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Abstract: Denote by $ \Delta _L^D$ the Laplacian on a hypercube in $ {{\mathbf{R}}^d}$ with side length $ \pi L$. Also denote by $ N\left( {\lambda ,A} \right)$ the number of eigenvalues of the operator $ A$ below $ \lambda $. If $ V \geq 0$ is a bounded function of compact support, ($ V > 0$ on a set of positive measure) then $ N\left( { - \Delta _L^D,\lambda } \right) - N\left( { - \Delta _L^D + V,\lambda } \right)$ is not bounded as $ L \to \infty $ for dimension $ d > 1$.

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  • [1] François Fricker, Einführung in die Gitterpunktlehre, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series], vol. 73, Birkhäuser Verlag, Basel-Boston, Mass., 1982 (German). MR 673938
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Keywords: Dirichlet Laplacian, small perturbation, eigenvalue distribution
Article copyright: © Copyright 1987 American Mathematical Society