Some results on the essential spectrum
There are very few results on the essential spectrum of the Laplacian on a noncompact complete manifold. In 1981, \textit{H. Donnelly} [Topology 20, 1-14 (1981; Zbl 0463.53027)] proved that the essential spectrum of a simply connected negatively curved Riemannian manifolds whose sectional curvature tends to \(-a\) at infinity is \([{1 \over 4} a(n-1)^ 2,+\infty)\); furthermore he and \textit{P. Li} [Duke Math. J. 46, 497-503 (1979; Zbl 0416.58025)] pointed out that under the same topological and geometric assumptions, if \(a = + \infty\), then its essential spectrum is an empty set. In this paper, we have the following theorems. Theorem 1. Suppose \(M\) is an \(n\)-dimensional noncompact complete Riemannian manifold with a pole and nonnegative Ricci curvature outside a compact set, then its essential spectrum is \([0,+\infty)\). Theorem 2. Suppose \(M\) is an \(n\)-dimensional noncompact complete Riemannian manifold with a rotational symmetric metric, and its radical curvature is bounded, then we have (1) the essential spectrum of the Laplacian is \([{1 \over 4} a^ 2,+ \infty)\), as \(\lim_{r \to \infty} \{\log V(r)-ar-b\}=0\), (2) the essential spectrum of the Laplacian is \([{1 \over 4} a^ 2,+ \infty)\), as \(\lim_{r \to \infty} \{\log (C-V(r)) + ar + b\} = 0\), where \(V(r)\) is the volume of the geodesic ball with center pole and radius \(r\); \(C\), \(a\) and \(b\) are positive constants.
scientific article; zbMATH DE number 563422