Perturbation expansions for eigenvalues and eigenvectors for a rectangular membrane subject to a restorative force. (English) Zbl 0887.35113

Summary: Series expansions are obtained for a rich subset of eigenvalues and eigenfunctions of an operator that arises in the study of rectangular membranes: the operator is the two-dimensional Laplacian with restorative force term and Dirichlet boundary conditions. Expansions are extracted by considering the restorative force term as a linear perturbation of the Laplacian; errors of truncation for these expansions are estimated. The criteria defining the subset of eigenvalues and eigenfunctions that can be studied depend only on the size and linearity of the perturbation. The results are valid for almost all rectangular domains.


35P20 Asymptotic distributions of eigenvalues in context of PDEs
35B20 Perturbations in context of PDEs
35C10 Series solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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