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The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Author: Zhongwei Shen
Journal: Proc. Amer. Math. Soc. 131 (2003), 3447-3456
MSC (2000): Primary 35P20, 35J10
Published electronically: February 20, 2003
MathSciNet review: 1990634
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Abstract: Let $M$ be a noncompact complete Riemannian manifold. We consider the Schrödinger operator $-\Delta +V$ acting on $L^{2}(M)$, where $V$ is a nonnegative, locally integrable function on $M$. We obtain some simple conditions which imply that $\inf \text{Spec} (-\Delta +V)$, the bottom of the spectrum of $-\Delta +V$, is strictly positive. We also establish upper and lower bounds for the counting function $N(\lambda )$.

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Additional Information

Zhongwei Shen
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Received by editor(s): May 27, 2002
Published electronically: February 20, 2003
Communicated by: Andreas Seeger
Article copyright: © Copyright 2003 American Mathematical Society