On the essential spectrum of the Laplacian on complete manifolds
The author proves two theorems on when the essential spectrum of the Laplace-Beltrami operator on a complete manifold \(M\) satisfying a generalized ``pole condition'' is equal to an interval \([c,\infty)\). The condition on \(M\) is that it contains an open set \(U\) with compact smooth boundary \(\partial U\) such that the outward normal exponential map is a diffeomorphism. The first theorem says that if the Ricci curvature evaluated in normal directions at distance \(t\) is bounded below by \(-(n-1)\phi(t)\) for some continuous function \(\phi\) vanishing at infinity, then the essential spectrum of \(M\) is equal to \([0,\infty)\). The second theorem states that if \(\Delta r(x)\) converges in a certain sense to a constant \(c\) at infinity, where \(r\) is the distance to \(U\), then the essential spectrum is equal to \([c^2/4,\infty)\).
scientific article; zbMATH DE number 1172181