Vol. 13, No. 6, 2020

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

Volume 16
Issue 7, 1485–1744
Issue 6, 1289–1483
Issue 5, 1089–1288
Issue 4, 891–1088
Issue 3, 613–890
Issue 2, 309–612
Issue 1, 1–308

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

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
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
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

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.

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