Absence of embedded eigenvalues for Riemannian Laplacians

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Abstract

In this paper we study absence of embedded eigenvalues for Schrödinger 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 is an upper bound of the trace of this quantity, while a third one is a bound of the derivatives of part of the trace (some oscillatory behaviour of the trace is allowed). In addition to geometric bounds we need conditions on the potential, a regularity property of the domain of the Schrödinger operator and the unique continuation property. Examples include ends endowed with asymptotic Euclidean or hyperbolic metrics. © 2013 Elsevier Inc.

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Ito, K., & Skibsted, E. (2013). Absence of embedded eigenvalues for Riemannian Laplacians. Advances in Mathematics, 248, 945–962. https://doi.org/10.1016/j.aim.2013.08.023

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