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Estimates for the eigenvalues of Hill's equation and applications for the eigenvalues of the laplacian on toroidal surfaces

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Abstract

In this paper we give upper and lower bounds for each eigenvalue λ n of Hill's differential equation. We apply the results to toroidal surfaces of revolution in order to get estimates for the eigenvalues of the Laplacian in terms of curvature expressions; they are sharp for the flat torus. As an example, we investigate the standard torus in IR3; here, the bounds depend on the radii only.

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Authors and Affiliations

  1. Riegerbusch 27, D-5800, Hagen

    Brigitte Beekmann

  2. Mathematisches Institut, Westfälische Wilhelms-Universität, Einsteinstraße 62, D-4400, Münster

    Hermann Lökes

Authors
  1. Brigitte Beekmann
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  2. Hermann Lökes
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Beekmann, B., Lökes, H. Estimates for the eigenvalues of Hill's equation and applications for the eigenvalues of the laplacian on toroidal surfaces. Manuscripta Math 68, 295–308 (1990). https://doi.org/10.1007/BF02568765

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  • DOI: https://doi.org/10.1007/BF02568765

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