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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.

MSC:

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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References:

[1] 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. · Zbl 0897.35058
[2] 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. · Zbl 0859.35136
[3] 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. · Zbl 0852.35142
[4] Joel Feldman, Horst Knörrer, and Eugene Trubowitz, The perturbatively stable spectrum of a periodic Schrödinger operator, Invent. Math. 100 (1990), no. 2, 259 – 300. · Zbl 0701.34082
[5] Leonid Friedlander, On the spectrum of the periodic problem for the Schrödinger operator, Comm. Partial Differential Equations 15 (1990), no. 11, 1631 – 1647. , https://doi.org/10.1080/03605309908820740 Leonid Friedlander, Erratum to: ”On the spectrum of the periodic problem for the Schrödinger operator”, Comm. Partial Differential Equations 16 (1991), no. 2-3, 527 – 529. · Zbl 0723.35055
[6] 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.
[7] 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.
[8] Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. · Zbl 0342.47009
[9] Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. · Zbl 0754.11020
[10] Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. · Zbl 0501.46003
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