The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Shen, Zhongwei

doi:10.1090/S0002-9939-03-06968-5

The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds
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Proc. Amer. Math. Soc. 131 (2003), 3447-3456 Request permission

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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