Eigenfunctions of the Laplace Operator on the Surface of a Triaxial Ellipsoid and in the Region Exterior to IT
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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 .
KeywordsLaplace Operator Asymptotic Expression Closed Geodesic Beltrami Operator Region Exterior
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