Vol. 13, No. 6, 2020

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 6, 1605–1954
Issue 5, 1269–1603
Issue 4, 945–1268
Issue 3, 627–944
Issue 2, 317–625
Issue 1, 1–316

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
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