Vol. 13, No. 6, 2020

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Eigenvalue bounds for non-self-adjoint Schrödinger operators with nontrapping metrics

Colin Guillarmou, Andrew Hassell and Katya Krupchyk

Vol. 13 (2020), No. 6, 1633–1670
Abstract

We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension n 3. Specifically, we are concerned with the following two types of estimates. The first one deals with Keller-type bounds on individual eigenvalues of the Schrödinger operator with a complex potential in terms of the Lp-norm of the potential, while the second one is a Lieb–Thirring-type bound controlling sums of powers of eigenvalues in terms of the Lp-norm of the potential. We extend the results of Frank (2011), Frank and Sabin (2017), and Frank and Simon (2017) on the Keller- and Lieb–Thirring-type bounds from the case of Euclidean spaces to that of nontrapping asymptotically conic manifolds. In particular, our results are valid for the operator Δg + V on n with g being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and V Lp complex-valued.

Keywords
non-self-adjoint Schrödinger operators, eigenvalue bounds, asymptotically conic manifolds
Mathematical Subject Classification 2010
Primary: 35P15, 42B37, 58J40, 58J50
Milestones
Received: 19 October 2017
Revised: 29 April 2019
Accepted: 13 August 2019
Published: 12 September 2020
Authors
Colin Guillarmou
Université Paris-Saclay, CNRS
Laboratoire de Mathématiques d’Orsay
Orsay
France
Andrew Hassell
Mathematical Sciences Institute
Australian National University
Canberra
Australia
Katya Krupchyk
Department of Mathematics
University of California
Irvine, CA
United States