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

Guillarmou, Colin; Hassell, Andrew; Krupchyk, Katya

doi:10.2140/apde.2020.13.1633

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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

DOI: 10.2140/apde.2020.13.1633

Abstract

We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension $n \geq 3$ . 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 $L^{p}$ -norm of the potential, while the second one is a Lieb–Thirring-type bound controlling sums of powers of eigenvalues in terms of the $L^{p}$ -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 $Δ_{g} + V$ on $ℝ^{n}$ with $g$ being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and $V \in L^{p}$ complex-valued.

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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

Vol. 13, No. 6, 2020