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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Genericity of simple eigenvalues for elliptic PDE's


Author: J. H. Albert
Journal: Proc. Amer. Math. Soc. 48 (1975), 413-418
MSC: Primary 58G15; Secondary 35P05
MathSciNet review: 0385934
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Abstract: The spectrum of a selfadjoint, $ {C^\infty }$ linear elliptic partial differential operator on a compact manifold contains only isolated eigenvalues, each having finite multiplicity. It is sometimes the case that these multiplicities are unbounded; this is common in problems arising in applications because of the high degree of symmetry usually present. The main theorem shows that the property of having only simple eigenvalues is generic for operators obtained by varying the zeroth order part of a given operator.


References [Enhancements On Off] (What's this?)

  • [1] J. H. Albert, Nodal and critical sets for eigenfunctions of elliptic operators, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1973, pp. 71–78. MR 0334291 (48 #12610)
  • [2] -, Topology of the nodal and critical point sets for eigenfunctions of elliptic, operators, Thesis, M. I. T., Cambridge, Mass., 1971.
  • [3] Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, American Mathematical Society, Providence, R.I., 1979. With supplements by Lars Gȧrding and A. N. Milgram; With a preface by A. S. Householder; Reprint of the 1964 original; Lectures in Applied Mathematics, 3A. MR 598466 (82c:35001)
  • [4] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391 (16,426a)
  • [5] Franz Rellich, Perturbation theory of eigenvalue problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0240668 (39 #2014)
  • [6] K. Uhlenbeck, Generic properties of eigenfunctions (manuscript).

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0385934-4
PII: S 0002-9939(1975)0385934-4
Keywords: Elliptic operator, eigenvalue, generic, perturbation
Article copyright: © Copyright 1975 American Mathematical Society