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## Perturbation expansions for eigenvalues and eigenvectors
for a rectangular membrane subject to a restorative force

### Joyce R. McLaughlin and Arturo Portnoy

**Abstract.**
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 2-D
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.

*Copyright 1997 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**03** (1997), pp. 72-77
- Publisher Identifier: S 1079-6762(97)00027-9
- 1991
*Mathematics Subject Classification*. Primary 35P20
*Key words and phrases*. Perturbation expansion, eigenvalue,
eigenvector, membrane, inverse nodal problem
- Received by the editors May 16, 1997
- Posted on August 19, 1997
- Communicated by Michael Taylor
- Comments (When Available)

**Joyce R. McLaughlin**

Rensselaer Polytechnic Institute, Troy, NY 12180

*E-mail address:* `mclauj@rpi.edu`

**Arturo Portnoy**

Rensselaer Polytechnic Institute, Troy, NY 12180

*E-mail address:* `portna@rpi.edu`

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