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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On oscillatory elliptic equations on manifolds

Authors: A. Baider and E. A. Feldman
Journal: Trans. Amer. Math. Soc. 258 (1980), 495-504
MSC: Primary 58G25; Secondary 35B05, 35J15
MathSciNet review: 558186
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Abstract: In this note we investigate the possibility of oscillatory behavior for a second-order selfadjoint elliptic operators on noncompact Riemannian manifolds (M, g). Let A be such an operator which is semibounded below and symmetric on $ C_0^\infty (M)\, \subseteq \,{L^2}(M,\,d\mu )$ where $ d\mu $ is a volume element on M. If $ \varphi $ is a $ {C^\infty }$ function such that $ A\varphi \, = \,\lambda \varphi $, we would naively say that $ \varphi $ is oscillatory (and by extension $ \lambda $ is oscillatory if it possesses such an eigenfunction $ \varphi $) if $ M\, - \,{\varphi ^{ - 1}}(0)$ has an infinite number of bounded connected components. For technical reasons this is not quite adequate for a definition. However, in §1 we give the usual definition of oscillatory which is a slight generalization of the one above. Let $ {\Lambda _0}$ be the number below which this phenomenon cannot occur; $ {\Lambda _0}$ is the oscillatory constant for the operator A. In that A is semibounded and symmetric on $ C_0^\infty (M)\, \subseteq \,{L^2}(M,\,d\mu )$, A has a Friedrichs extension. Let $ {\Lambda _c}$ be the bottom of the continuous spectrum of the Friedrichs extension of A. Our main result is $ {\Lambda _0}\, = \,{\Lambda _c}$.

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Article copyright: © Copyright 1980 American Mathematical Society