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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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