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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Genericity of simple eigenvalues for elliptic PDE’s
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by J. H. Albert PDF
Proc. Amer. Math. Soc. 48 (1975), 413-418 Request permission

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
  • 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
  • —, Topology of the nodal and critical point sets for eigenfunctions of elliptic, operators, Thesis, M. I. T., Cambridge, Mass., 1971.
  • Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, Lectures in Applied Mathematics, vol. 3, 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. MR 598466
  • R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • Franz Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Assisted by J. Berkowitz; With a preface by Jacob T. Schwartz. MR 0240668
  • K. Uhlenbeck, Generic properties of eigenfunctions (manuscript).
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 48 (1975), 413-418
  • MSC: Primary 58G15; Secondary 35P05
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0385934-4
  • MathSciNet review: 0385934