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

McLaughlin, Joyce; Portnoy, Arturo

doi:10.1090/S1079-6762-97-00027-9

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
Full-text PDF Free Access

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.

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.

Ole H. Hald and Joyce R. McLaughlin, Inverse nodal problems: finding the potential from nodal lines, Mem. Amer. Math. Soc. 119 (1996), no. 572, viii+148. MR 1370425, DOI https://doi.org/10.1090/memo/0572

Joyce R. McLaughlin and Ole H. Hald, A formula for finding a potential from nodal lines, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 2, 241–247. MR 1302784, DOI https://doi.org/10.1090/S0273-0979-1995-00584-7

Joel Feldman, Horst Knörrer, and Eugene Trubowitz, The perturbatively stable spectrum of a periodic Schrödinger operator, Invent. Math. 100 (1990), no. 2, 259–300. MR 1047135, DOI https://doi.org/10.1007/BF01231187

Leonid Friedlander, On the spectrum of the periodic problem for the Schrödinger operator, Comm. Partial Differential Equations 15 (1990), no. 11, 1631–1647. MR 1079606, DOI https://doi.org/10.1080/03605309908820740

Yu. E. Karpeshina, Analytic perturbation theory for a periodic potential, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 45–65 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 43–64. MR 992978, DOI https://doi.org/10.1070/IM1990v034n01ABEH000584

Yu. E. Karpeshina, Geometrical background for the perturbation theory of the polyharmonic operator with periodic potentials, Topological phases in quantum theory (Dubna, 1988) World Sci. Publ., Teaneck, NJ, 1989, pp. 251–276. MR 1126533

Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617

Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. MR 1176315

Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 564653