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Construction and spectrum
of the Anderson Hamiltonian with white noise potential on
$\mathbb R^2$ and $\mathbb R^3$
Yueh-Sheng Hsu and Cyril Labbé |
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Vol. 7 (2026), No. 1, 1–35
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DOI: 10.2140/pmp.2026.7.1
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Abstract |
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We propose a simple construction of the Anderson Hamiltonian with white noise potential on and based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau (1981) that associates a unique self-adjoint generator to a symmetric semigroup satisfying some mild assumptions. Then, we show that almost surely the spectrum of this random Schrödinger operator is . To prove this result, we extend the method of Kotani (1985) to our setting of singular random operators. |
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Keywords
Anderson Hamiltonian, white noise, random Schrödinger
operator, regularity structures, parabolic Anderson model,
spectrum, selfadjointness
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Mathematical Subject Classification
Primary: 35J10, 60H15
Secondary: 47A10
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Milestones
Received: 6 February 2024
Revised: 25 June 2025
Accepted: 13 October 2025
Published: 21 November 2025
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Authors |
| Yueh-Sheng Hsu | |
| Institute of Analysis and Scientific
Computing Technische Universität Wien 1040 Vienna Austria |
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| Cyril Labbé | |
| Laboratoire de Probabilités,
Statistique et Modélisation, UMR 8001 Université Paris Cité 75205 Paris France |
Institut Universitaire de France |
| © 2026 MSP (Mathematical Sciences Publishers). |
