Anderson localization for the $1$-d Schrödinger operator with white noise potential

Laure Dumaz; Cyril Labbé

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arXiv · 2212.04862

Anderson localization for the $1$-d Schrödinger operator with white noise potential

Laure Dumaz·Cyril Labbé

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Abstract

We consider the random Schrödinger operator on $\mathbb{R}$ obtained by perturbing the Laplacian with a white noise. We prove that Anderson localization holds for this operator: almost surely the spectral measure is pure point and the eigenfunctions are exponentially localized. We give two separate proofs of this result. We also present a detailed construction of the operator and relate it to the parabolic Anderson model. Finally, we discuss the case where the noise is smoothed out.

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Laure Dumaz, Cyril Labbé. 2022-12-09. Anderson localization for the $1$-d Schrödinger operator with white noise potential. https://arxiv.org/abs/2212.04862

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