A Continuum Version of the Kunz-Souillard Approach to Localization in One Dimension

TL;DR

This paper extends the Kunz-Souillard approach to continuum one-dimensional Schrödinger operators with random potentials, proving exponential decay of correlators and establishing localization uniformly across energy regions.

Contribution

It introduces a continuum analog of the Kunz-Souillard method, providing a new proof of localization for continuum 1D Schrödinger operators with Anderson-type potentials.

Findings

Proves exponential decay of finite volume correlators

Establishes dynamical and spectral localization

Uniform results across energy regions

Abstract

We consider continuum one-dimensional Schr\"odinger operators with potentials that are given by a sum of a suitable background potential and an Anderson-type potential whose single-site distribution has a continuous and compactly supported density. We prove exponential decay of the expectation of the finite volume correlators, uniform in any compact energy region, and deduce from this dynamical and spectral localization. The proofs implement a continuum analog of the method Kunz and Souillard developed in 1980 to study discrete one-dimensional Schr\"odinger operators with potentials of the form background plus random.