Negative curvature and embedded eigenvalues
type: Item
instance of: scholarly article
title: Negative curvature and embedded eigenvalues
zbMATH Open document ID: 0699.53052
DOI: 10.1007/BF02570738
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174129
review text: 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)].
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 4147029
zbMATH Keywords: negative sectional curvature
zbMATH Keywords: essential spectrum
zbMATH Keywords: Laplacian
zbMATH Keywords: decay conditions
author: Harold Donnelly
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4147029