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

TL;DR

This paper proves Anderson localization for a one-dimensional Schrödinger operator with white noise potential, showing pure point spectrum and exponential localization of eigenfunctions, with two different proofs and discussion on noise smoothing.

Contribution

It provides the first rigorous proof of Anderson localization for the 1D Schrödinger operator with white noise potential, including detailed construction and relation to the parabolic Anderson model.

Findings

Spectral measure is pure point almost surely.

Eigenfunctions are exponentially localized.

Two independent proofs of localization.

Abstract

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.