Genericity of simple eigenvalues for elliptic PDE's
J. H. Albert
Proc. Amer. Math. Soc. 48 (1975), 413-418
Primary 58G15; Secondary 35P05
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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.
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.
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
Courant and D.
Hilbert, Methods of mathematical physics. Vol. I, Interscience
Publishers, Inc., New York, N.Y., 1953. MR 0065391
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
K. Uhlenbeck, Generic properties of eigenfunctions (manuscript).
- J. H. Albert, Nodal and critical sets for eigenfunctions of elliptic operators, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, R. I., 1973, pp. 71-78. MR 0334291 (48:12610)
- -, Topology of the nodal and critical point sets for eigenfunctions of elliptic, operators, Thesis, M. I. T., Cambridge, Mass., 1971.
- L. Bers, F. John and M. Schechter, Partial differential equations, Lectures in Appl. Math., vol. 3, Interscience, New York, 1964. MR 29 #346. MR 598466 (82c:35001)
- R. Courant and D. Hilbert, Methods of mathematical physics. Vols. I, II, Interscience, New York, 1953, 1962. MR 16, 426; 25 #4216. MR 0065391 (16:426a)
- F. Rellich, Perturbation theory of eigenvalue problems, Gordon and Breach, New York, 1969. MR 39 #2014. MR 0240668 (39:2014)
- K. Uhlenbeck, Generic properties of eigenfunctions (manuscript).
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