On oscillatory elliptic equations on manifolds

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 ${\Lambda _0} = {\Lambda _c}$.