Moment Analysis for Localization in Random Schroedinger Operators

TL;DR

This paper advances the understanding of localization in random Schrödinger operators by establishing exponential decay bounds on transition amplitudes using fractional moments, overcoming previous technical challenges in the continuum case.

Contribution

It introduces a novel approach employing weak-L1 estimates and resolvent boundary-value distributions to extend fractional moment methods to continuum Schrödinger operators.

Findings

Exponential decay bounds on transition amplitudes and projection kernels.

Finite fractional moments of the resolvent due to disorder effects.

Overcoming technical barriers in continuum localization analysis.

Abstract

We study localization effects of disorder on the spectral and dynamical properties of Schroedinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.