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Random Schrödinger
operators with complex decaying potentials
Jean-Claude Cuenin and Konstantin Merz |
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Vol. 18 (2025), No. 2, 279–306
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Abstract |
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We prove that the eigenvalues of a continuum random Schrödinger operator of Anderson-type, with complex decaying potential, can be bounded (with high probability) in terms of an norm of the potential for all . This shows that, in the random setting, the exponent can be essentially doubled compared to the deterministic bounds of Frank (Bull. Lond. Math. Soc. 43:4 (2011), 745–750). This improvement is based on ideas of Bourgain (Discrete Contin. Dyn. Syst. 8:1, (2002), 1–15) related to almost-sure scattering for lattice Schrödinger operators. |
Keywords
random Schrödinger operator, long-range potential, complex
eigenvalue estimates, Lieb–Thirring
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Mathematical Subject Classification
Primary: 35P15
Secondary: 35R60, 81Q12, 82B44
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Milestones
Received: 29 March 2022
Revised: 6 March 2023
Accepted: 22 October 2023
Published: 5 February 2025
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Authors |
| Jean-Claude Cuenin | |
| Department of Mathematical
Sciences Loughborough University Loughborough United Kingdom |
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| Konstantin Merz | |
| Institut für Analysis und
Algebra Technische Universität Braunschweig Braunschweig Germany |
Department of Mathematics Graduate School of Science Osaka University Osaka Japan |
| © 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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