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Eigenvalue bounds for
non-self-adjoint Schrödinger operators with nontrapping
metrics
Colin Guillarmou, Andrew Hassell and Katya Krupchyk |
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Vol. 13 (2020), No. 6, 1633–1670
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Abstract |
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We study eigenvalues of non-self-adjoint Schrödinger operators on nontrapping asymptotically conic manifolds of dimension . 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 -norm of the potential, while the second one is a Lieb–Thirring-type bound controlling sums of powers of eigenvalues in terms of the -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 on with being a nontrapping compactly supported (or suitably short-range) perturbation of the Euclidean metric and complex-valued. |
Keywords
non-self-adjoint Schrödinger operators, eigenvalue bounds,
asymptotically conic manifolds
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Mathematical Subject Classification 2010
Primary: 35P15, 42B37, 58J40, 58J50
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Milestones
Received: 19 October 2017
Revised: 29 April 2019
Accepted: 13 August 2019
Published: 12 September 2020
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Authors |
| Colin Guillarmou | |
| Université Paris-Saclay, CNRS Laboratoire de Mathématiques d’Orsay Orsay France |
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| Andrew Hassell | |
| Mathematical Sciences
Institute Australian National University Canberra Australia |
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| Katya Krupchyk | |
| Department of Mathematics University of California Irvine, CA United States |
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