Eigenfunctions of the Laplace Operator on the Surface of a Triaxial Ellipsoid and in the Region Exterior to IT
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  (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 , The complete agreement of the results confirms the conclusions of .
Unable to display preview. Download preview PDF.
- 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.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.Morse, P. M., and Feshbach, H, Methods of Theoretical Physics, McGraw-Hill, New York (1953).Google Scholar
- 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.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.Vainshtein, L. A., Ray flows on the triaxial ellipsoid, in: High-Power Electronics [in Russian], Vol. 4, Moscow (1965).Google Scholar