Boosted Simon‐Wolff Spectral Criterion and Resonant Delocalization

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 Schrödinger 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.

Multiplicity bound of singular spectrum for higher rank Anderson models

“…In the Anderson tight binding model, which is the rank one case, Barry Simon [16] showed that any standard basis vector δ n is cyclic in region of pure point spectrum. Other works in pure point regime are by Klein-Molchanov [10] and Aizenman-Warzel [2]. Jakšić-Last in [7,9] showed that the singular spectrum is almost surely simple in case of Anderson type Hamiltonians where rank of the perturbation is one.…”