Negative curvature and embedded eigenvalues
Let M be a complete simply connected Riemannian manifold of negative sectional curvature K. If \(K\to -1\) at infinity then the essential spectrum of the Laplacian consists of the half line \([(n-1)^ 2/4,\infty)\) [the author, Topology 20, 1-14 (1981; Zbl 0463.53027)]. The author proves that if \(K+1\) satisfies certain natural decay conditions then there is no square integrable eigenfunction on M with eigenvalue \(\lambda >(n-1)^ 2/4\) (embedded eigenvalue) in contrast to the existence of arbitrarily many eigenvalues \(\lambda <(n-1)^ 2/4,\) in general. The present paper removes all the unappealing hypotheses on derivatives of \(K+1\) in a previous paper of the author on the same theme [the author, Mich. Math. J. 28, 93-62 (1981; Zbl 0463.53028)].
scientific article; zbMATH DE number 4147029