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arXiv · 1109.1928

Absence of embedded eigenvalues for Riemannian Laplacians

K. Ito·E. Skibsted
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Abstract

In this paper we study absence of embedded eigenvalues for Schr\"odinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end. Another condition may be viewed (at least in a special case) as being a bound of the trace of this quantity, while similarly, a third one as being a bound of the derivative of this trace. In addition to geometric bounds we need conditions on the potential, a regularity property of the domain of the Schr\"odinger operator and the unique continuation property. Examples include ends endowed with asymptotic Euclidean or hyperbolic metrics studied previously in the literature.

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K. Ito, E. Skibsted. 2011-09-09. Absence of embedded eigenvalues for Riemannian Laplacians. https://arxiv.org/abs/1109.1928

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