Advertisement

Eigenfunctions of the Laplace Operator on the Surface of a Triaxial Ellipsoid and in the Region Exterior to IT

  • T. F. Pankratova
Part of the Seminars in Mathematics book series (SM, volume 9)

Abstract

The present paper deals with the construction of asymptotic expressions for quasi-eigenvalues and -eigenfunctions of the Laplace operator in the region exterior to a triaxial ellipsoid and the eigen-functions of the Laplace operator on the surface of the ellipsoid. Quasi-eigenvalues and -eigenfunctions and also eigenvalues and eigenfunctions mean the same as in [1] (a condensed review of these questions is found in §1 of the present paper). Quasi-eigenvalues are defined for functions concentrated at the surface of the ellipsoid. Asymptotic expressions are found for those eigenfunctions which differ appreciably from zero only in a neighborhood of the principal ellipses of the ellipsoid. The asymptotic expressions for the eigenfunctions of the Laplace operator on the surface of the ellipsoid found by the standard-equation method are compared with the expressions found by the parabolic-equation method in [2], The complete agreement of the results confirms the conclusions of [2].

Keywords

Laplace Operator Asymptotic Expression Closed Geodesic Beltrami Operator Region Exterior 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    Babich, V. M., The asymptotic behavior of quasi-eigenvalues of the exterior problem for the Laplace operator, in: Topics in Mathematical Physics, Vol. 2, M. Sh. Birman, ed., Consultants Bureau, New York (1968).Google Scholar
  2. 2.
    Babich, V. M., and Lazutkin, V. F., Eigenfunctions concentrated near a closed geodesic, in: Topics in Mathematical Physics, Vol. 2, M. Sh. Birman, ed., Consultants Bureau, New York (1968).Google Scholar
  3. 3.
    Morse, P. M., and Feshbach, H, Methods of Theoretical Physics, McGraw-Hill, New York (1953).Google Scholar
  4. 4.
    Marchenko, V. A., and Khruslov, E. Ya., Analytic properties of the resolvent of a certain boundary value problem, Third All-Union Symposium on Wave Diffraction, Tblisi, 24–30 September, 1964, Report [in Russian], Nauka.Google Scholar
  5. 5.
    Bykov, Vc P., The geometric optics of three-dimensional oscillations in open resonators, in: High-Power Electronics [in Russian], Vol. 4, Moscow (1965).Google Scholar
  6. 6.
    Vainshtein, L. A., Ray flows on the triaxial ellipsoid, in: High-Power Electronics [in Russian], Vol. 4, Moscow (1965).Google Scholar

Copyright information

© Consultants Bureau, New York 1970

Authors and Affiliations

  • T. F. Pankratova

There are no affiliations available

Personalised recommendations