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Transactions of the American Mathematical Society

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Generic properties of eigenfunctions of elliptic partial differential operators


Author: Jeffrey H. Albert
Journal: Trans. Amer. Math. Soc. 238 (1978), 341-354
MSC: Primary 58G99; Secondary 35J15, 35P99
DOI: https://doi.org/10.1090/S0002-9947-1978-0471000-3
MathSciNet review: 0471000
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Abstract: The problem considered here is that of describing generically the zeros, critical points and critical values of eigenfunctions of elliptic partial differential operators. We consider operators of the form $ L + \rho $, where L is a fixed, second-order, selfadjoint, $ {C^\infty }$ linear elliptic partial differential operator on a compact manifold (without boundary) and $ \rho $ is a $ {C^\infty }$ function. It is shown that, for almost all $ \rho $, i.e. for a residual set, the eigenvalues of $ L + \rho $ are simple and the eigenfunctions have the following properties: (1) they are Morse functions; (2) distinct critical points have distinct critical values; (3) 0 is not a critical value.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0471000-3
Article copyright: © Copyright 1978 American Mathematical Society

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