Noncompact Riemannian manifolds with purely continuous spectrum (Q1066477)

Let M be a complete simply connected Riemannian manifold of non-positive sectional curvature. The exponential map is a diffeomorphism, so one introduces coordinates (r,\(\theta)\) as generalized polar coordinates. Let \(\Delta\) be the Laplacian and let K(r,\(\theta)\) be the curvature. Assume \(K\to 0\) at infinity and that certain other technical hypotheses are satisfied. Then \(\Delta\) has no point spectrum. In particular, if the metric is radially symmetric about some point on a surface and if \(K\geq - 1/(r^ 2 Log(r))\), then the spectrum of \(\Delta\) is purely continuous. This generalizes results of Donnelly and Pinsky.