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Exhaustion functions and the spectrum of Riemannian manifolds. (English) Zbl 0909.58055
Suppose that $$M$$ is a complete, noncompact, Riemannian manifold, which admits an exhaustion function $$b$$ satisfying certain asymptotic geometric hypotheses. If these hypotheses hold in an $$L^2$$-sense, then it is proved that the essential spectrum of $$-\Delta$$ is $$[0,\infty)$$. Here $$\Delta$$ denotes the Laplacian associated to the Riemannian metric of $$M$$. If $$b$$ satisfies stronger pointwise hypotheses, then it is shown that $$-\Delta$$ has no positive eigenvalues. The model case is $$M= \mathbb{R}^n$$, with its standard flat metric, and $$b$$ the Euclidean distance from the origin.

MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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