#### 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 $n\ge 3$. 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 ${\Delta }_{g}+V$ 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.

##### 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