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Pré-Publication, Document De Travail Année : 2022

Anderson localization for the $1$-d Schr\"odinger operator with white noise potential

Résumé

We consider the random Schr\"odinger 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.

Dates et versions

hal-03897532 , version 1 (13-12-2022)

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Citer

Laure Dumaz, Cyril Labbé. Anderson localization for the $1$-d Schr\"odinger operator with white noise potential. 2022. ⟨hal-03897532⟩
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