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Quasiclassical Methods pp 115–124Cite as

On the Asymptotic Distribution of Eigenvalues in Gaps

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 95))

Abstract

Virtually all results on eigenvalue asymptotics for differential operators have their roots in Weyl’s celebrated law for the distribution of the eigenvalues

$$ 0 < E_1 < E_2 \leqslant E_3 \leqslant \ldots ,E_k \to \infty {\text{ }}as{\text{ }}k \to \infty , $$

of the Dirichlet Laplacian -△ on an open, bounded domain Ω ⊂ Rm: If N(λ) denotes the number of eigenvalues E k < λ, then

$$ N\left( \lambda \right) \sim c_d vol\left( \Omega \right)\lambda ^{m/2} ,\lambda \to \infty , $$
((1))

under mild regularity assumptions on the boundary δΩ here c m is a universal constant which depends only on the dimension m.

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Author information

Authors and Affiliations

  1. Institut für Analysis, TU Braunschweig, Pockelsstr. 14, D-38106, Braunschweig, Germany

    Rainer Hempel

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  1. Rainer Hempel
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1003, USA

    Jeffrey Rauch

  2. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125-0001, USA

    Barry Simon

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© 1997 Springer Science+Business Media New York

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Hempel, R. (1997). On the Asymptotic Distribution of Eigenvalues in Gaps. In: Rauch, J., Simon, B. (eds) Quasiclassical Methods. The IMA Volumes in Mathematics and its Applications, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1940-8_5

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  • DOI: https://doi.org/10.1007/978-1-4612-1940-8_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7349-3

  • Online ISBN: 978-1-4612-1940-8

  • eBook Packages: Springer Book Archive

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