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

Authors:
Joyce R. McLaughlin and Arturo Portnoy

Journal:
Electron. Res. Announc. Amer. Math. Soc. **3** (1997), 72-77

MSC (1991):
Primary 35P20

DOI:
https://doi.org/10.1090/S1079-6762-97-00027-9

Published electronically:
August 19, 1997

MathSciNet review:
1461976

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Abstract | References | Similar Articles | Additional Information

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.

**[MP]**Joyce R. McLaughlin and Arturo Portnoy,*Perturbing a rectangular membrane with a restorative force: effects on eigenvalues and eigenfunctions*, to appear in Communications in Partial Differential Equations.**[HM]**Ole H. Hald and Joyce R. McLaughlin,*Inverse nodal problems: finding the potential from nodal lines*, Memoirs of the American Mathematical Society, January 1996, Volume 119, Number 572. MR**97d:35240****[MH]**Joyce R. McLaughlin and Ole H. Hald,*A formula for finding a potential from nodal lines*, Bulletin (new series) of the American Mathematical Society (1995), Volume 32, Number 2, pp. 241. MR**95g:35213****[FKT]**J. Feldman, H. Knörrer and E. Trubowitz,*The perturbatively stable spectrum of a periodic Schrödinger operator*, Inventiones Matematicae, Vol. 100 (1990), pp. 72-77. MR**91m:35167****[F]**L. Friedlander,*On the spectrum of the periodic problem for the Schrödinger operator*, Comm. P.D.E., Vol. 15 (1990), pp. 1631-1647. MR**92i:35092a****[KP]**Yu. E. Karpeshina,*Analytic perturbation theory for a periodic potential*, Math. USSR Izvestiya, Vol. 34 (1990), No.1, pp. 43-64. MR**91f:47065****[KP2]**Yu. E. Karpeshina,*Geometrical background for the perturbation theory of the polyharmonic operator with periodic potentials*, International Seminar on Geometrical Aspects of Quantum Theory, Topological Phases in Quantum Theory, World Scientific, 1988, pp. 251-276. MR**93c:81044****[K]**Tosio Kato,*Perturbation theory for linear operators*, New York: Springer-Verlag, 2nd Ed., 1976. MR**53:11389****[S]**Wolfgang M. Schmidt,*Diophantine approximations and Diophantine equations*, New York: Springer Verlag, 1991. MR**94f:11059****[TL]**Angus E. Taylor and David C. Lay,*Introduction to functional analysis*, New York: Wiley, 1980. MR**81b:46001**

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Additional Information

**Joyce R. McLaughlin**

Affiliation:
Rensselaer Polytechnic Institute, Troy, NY 12180

Email:
mclauj@rpi.edu

**Arturo Portnoy**

Affiliation:
Rensselaer Polytechnic Institute, Troy, NY 12180

Email:
portna@rpi.edu

DOI:
https://doi.org/10.1090/S1079-6762-97-00027-9

Keywords:
Perturbation expansion,
eigenvalue,
eigenvector,
membrane,
inverse nodal problem

Received by editor(s):
May 16, 1997

Published electronically:
August 19, 1997

Communicated by:
Michael Taylor

Article copyright:
© Copyright 1997
American Mathematical Society