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

2022/12/09 by Laure Dumaz, Cyril Labbé, Dumaz, Laure +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Probability (math.PR) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Statistical Mechanics (cond-mat.stat-mech)

paper · pdf · doi:10.48550/arxiv.2212.04862

openalex publication_date 2022/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the random Schrödinger operator on 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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