Absence of embedded eigenvalues for Riemannian Laplacians

TL;DR

This paper investigates conditions under which Schrödinger operators on non-compact Riemannian manifolds lack embedded eigenvalues, focusing on geometric bounds and potential conditions, with applications to Euclidean and hyperbolic ends.

Contribution

It establishes new geometric and potential conditions ensuring the absence of embedded eigenvalues for Riemannian Laplacians, extending previous results to more general manifolds.

Findings

Embedded eigenvalues are absent under specified geometric bounds.

Conditions on the potential and domain regularity are crucial.

Applications include manifolds with Euclidean or hyperbolic ends.

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.