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arXiv · 2401.17900

Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$

Yueh-Sheng Hsu·Cyril Labbé
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Abstract

We propose a simple construction of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$ based on the solution theory of the parabolic Anderson model. It relies on a theorem of Klein and Landau [KL81] 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 $\mathbf{R}$. To prove this result, we extend the method of Kotani [Kot85] to our setting of singular random operators.

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Yueh-Sheng Hsu, Cyril Labbé. 2025-07-10. Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$. https://doi.org/10.2140/pmp.2026.7.1

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