On Oscillatory Elliptic Equations on Manifolds

• Published in: Transactions of the American Mathematical Society Vol. 258; no. 2; pp. 495 - 504
• Main Authors: Baider, A., Feldman, E. A.
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 1980
• Subjects: Continuous functions; Continuous spectra; Elliptic equations; Mathematical functions; Mathematical theorems; Riemann manifold; Symmetry; Topological theorems; Vertices
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-1980-0558186-6

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 $C^\infty_0(M) \subseteq L^2(M, d\mu)$ where $d\mu$ is a volume element $M$. 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 possessess such an eigenfunctio $\varphi$) 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 $\S1$ 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 $C^\infty_0(M) \subseteq L^2(M, d\mu), A$ has a Friedrichs extension. Let $\Lambda be the bottom of the continuous spectrum has a Fredrichs extension of $A$. Our main result is $\Lambda_0 = \Lambda_c$.