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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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  • [1] A. Baider, Non-compact Riemannian manifolds with discrete spectra, J. Differential Geometry (to appear).
  • [2] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1971. MR 0282313 (43:8025)
  • [3] I. Chavel and E. Feldman, Spectra of domains, J. Functional Analysis 30 (1978), 198-222. MR 515225 (80c:58027)
  • [4] S. Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), 43-55. MR 0397805 (53:1661)
  • [5] R. Courant and D. Hilbert, Methods of mathematical physics, vol. I, Interscience, New York, 1952. MR 0065391 (16:426a)
  • [6] P. Hartman and C. R. Putnam, The least cluster point of the spectrum of boundary value problems, Amer. J. Math. 70 (1968), 849-855. MR 0027928 (10:376f)
  • [7] J. Piepenbrink, Nonoscillatory elliptic equations, J. Differential Equations 15 (1974), 541-550. MR 0342829 (49:7573)
  • [8] J. Rauch, Partial differential equations and related topics, Lecture Notes in Math., vol. 446, Springer-Verlag, Berlin and New York, 1975, pp. 354-389.
  • [9] J. Rauch and M. Taylor, Potential and scattering theory on wildly perturbed domains, J. Functional Analysis 18 (1975), 27-59. MR 0377303 (51:13476)
  • [10] F. Riesz and B. Nagy, Functional analysis, Ungar, New York, 1955.
  • [11] D. Stone, Geodesies in piecewise manifolds, Trans. Amer. Math. Soc. 215 (1976), 1-44. MR 0402648 (53:6464)
  • [12] H. Whitney, Geometric integration theory, Princeton Mathematical Series, vol. 21, Princeton Univ. Press, Princeton, N. J., 1957. MR 0087148 (19:309c)

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Article copyright: © Copyright 1980 American Mathematical Society

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