Expansion of the almost sure spectrum in the weak disorder regime

TL;DR

This paper investigates how the spectrum of random ergodic Schrödinger operators expands at its bottom when weak disorder is introduced, providing estimates on the spectrum's growth in the perturbative regime.

Contribution

It offers new quantitative estimates on the spectrum expansion at the bottom for operators on ^2( Z^d) under weak disorder conditions.

Findings

Spectrum expands at the bottom with increasing disorder.

Quantitative bounds on the spectrum's expansion are derived.

Results apply to operators on ^2( Z^d).

Abstract

The spectrum of random ergodic Schr\"odinger-type operators is almost surely a deterministic subset of the real line. The random operator can be considered as a perturbation of a periodic one. As soon as the disorder is switched on via a global coupling constant, the spectrum expands. We estimate how much the spectrum expands at its bottom for operators on $\ell^2(\mathbb Z^d)$.