# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## On oscillatory elliptic equations on manifoldsHTML articles powered by AMS MathViewer

by A. Baider and E. A. Feldman
Trans. Amer. Math. Soc. 258 (1980), 495-504 Request permission

## 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 ${\Lambda _0} = {\Lambda _c}$.
References
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G25, 35B05, 35J15
• Retrieve articles in all journals with MSC: 58G25, 35B05, 35J15