Boosted Simon-Wolff Spectral Criterion and Resonant Delocalization

TL;DR

This paper enhances the Simon-Wolff spectral criterion to better identify continuous spectra in random operators, applying it to Schrödinger operators and proving spectrum simplicity under certain conditions.

Contribution

It introduces a measurable at infinity condition for the Simon-Wolff criterion, boosts it with a zero-one law, and applies it to establish continuous spectrum and spectrum simplicity results.

Findings

Enhanced Simon-Wolff criterion for continuous spectrum detection

Proved almost sure simplicity of pure point spectrum under certain conditions

Applied boosted criterion to random Schrödinger operators with new sufficiency conditions

Abstract

Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon-Wolff criterion is shown to be measurable at infinity. By implication, for the iid case and more generally potentials with the K-property the criterion is boosted by a zero-one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schr\"odinger operators. The general proof strategy which this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon-Wolff rank-one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.