Grand Ensembles of deterministic operators. II. Localization for generic `haarsh' potentials

TL;DR

This paper proves exponential dynamical localization for a class of lattice Schrödinger operators with complex, discontinuous deterministic potentials generated by Haar-like expansions, using a multi-scale analysis approach.

Contribution

It introduces a novel method to establish localization for operators with 'haarsh' potentials depending on infinite parameters, expanding the understanding of deterministic localization phenomena.

Findings

Proves exponential dynamical localization in the strong disorder regime.

Develops a variant of Multi-Scale Analysis suited for 'haarsh' potentials.

Demonstrates localization for a new class of deterministic operators.

Abstract

We consider a particular class of lattice Schr\"odinger operators with deterministic potentials depending upon an infinite number of parameters in an auxiliary measurable space. We prove exponential dynamical localization for generic families in the strong disorder regime, using a variant of the Multi-Scale Analysis. In our model, the potential is generated by a function on a torus which is discontinuous (`haarsh') and constructed with the help of an expansion which reminds Haar's wavelet expansions.