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), 7277
MSC (1991):
Primary 35P20
Published electronically:
August 19, 1997
MathSciNet review:
1461976
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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 2D 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, Mem. Amer. Math. Soc. 119 (1996),
no. 572, viii+148. MR 1370425
(97d:35240)
 [MH]
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
(95g:35213), http://dx.doi.org/10.1090/S027309791995005847
 [FKT]
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
(91m:35167), http://dx.doi.org/10.1007/BF01231187
 [F]
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
(92i:35092a), http://dx.doi.org/10.1080/03605309908820740
 [KP]
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. USSRIzv.
34 (1990), no. 1, 43–64. MR 992978
(91f:47065)
 [KP2]
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
(93c:81044)
 [K]
Tosio
Kato, Perturbation theory for linear operators, 2nd ed.,
SpringerVerlag, Berlin, 1976. Grundlehren der Mathematischen
Wissenschaften, Band 132. MR 0407617
(53 #11389)
 [S]
Wolfgang
M. Schmidt, Diophantine approximations and Diophantine
equations, Lecture Notes in Mathematics, vol. 1467,
SpringerVerlag, Berlin, 1991. MR 1176315
(94f:11059)
 [TL]
Angus
Ellis Taylor and David
C. Lay, Introduction to functional analysis, 2nd ed., John
Wiley & Sons, New YorkChichesterBrisbane, 1980. MR 564653
(81b:46001)
 [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. 7277. 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. 16311647. MR 92i:35092a
 [KP]
 Yu. E. Karpeshina, Analytic perturbation theory for a periodic potential, Math. USSR Izvestiya, Vol. 34 (1990), No.1, pp. 4364. 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. 251276. MR 93c:81044
 [K]
 Tosio Kato, Perturbation theory for linear operators, New York: SpringerVerlag, 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:
http://dx.doi.org/10.1090/S1079676297000279
PII:
S 10796762(97)000279
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
