Fractional Moment Methods for Anderson Localization in the Continuum

TL;DR

This paper extends the fractional moment method to continuum random Schrödinger operators, providing new bounds on transition amplitudes and overcoming previous spectral analysis obstacles.

Contribution

It introduces a novel extension of the fractional moment method to continuum operators, with new exponential decay bounds and a solution to spectral shift bounds.

Findings

Exponential decay bounds for transition amplitudes in the continuum

Extension of fractional moment method to continuum operators

Resolution of spectral shift bound obstacle

Abstract

The fractional moment method, which was initially developed in the discrete context for the analysis of the localization properties of lattice random operators, is extended to apply to random Schr\"odinger operators in the continuum. One of the new results for continuum operators are exponentially decaying bounds for the mean value of transition amplitudes, for energies throughout the localization regime. An obstacle which up to now prevented an extension of this method to the continuum is the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. This difficulty is resolved through an analysis of the resonance-diffusing effects of the disorder.