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

DOI:
https://doi.org/10.1090/S0002-9939-1975-0385934-4

MathSciNet review:
0385934

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Abstract: The spectrum of a selfadjoint, 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.

**[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****[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 GÈ§rding and A. N. Milgram; With a preface by A. S. Householder; Reprint of the 1964 original; Lectures in Applied Mathematics, 3A. MR**598466****[4]**R. Courant and D. Hilbert,*Methods of mathematical physics. Vol. I*, Interscience Publishers, Inc., New York, N.Y., 1953. MR**0065391****[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****[6]**K. Uhlenbeck,*Generic properties of eigenfunctions*(manuscript).

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0385934-4

Keywords:
Elliptic operator,
eigenvalue,
generic,
perturbation

Article copyright:
© Copyright 1975
American Mathematical Society