Anderson localization for 1-d quasi-periodic Schr\"odinger operators with degenerate weights

TL;DR

This paper proves Anderson localization for one-dimensional quasi-periodic Schr"odinger operators with degenerate weights, expanding understanding in mathematical physics models with analytic, quasi-periodic potentials and weights.

Contribution

It introduces new techniques to establish localization in operators with degenerate weights, a case not previously well-understood in the literature.

Findings

Proves Anderson localization for 1D quasi-periodic Schr"odinger operators with degenerate weights.

Applicable to models like the Frenkel-Kontorova with impurities and singular Sturm-Liouville discretizations.

Extends localization results to a broader class of physical and mathematical models.

Abstract

We establish Anderson localization for 1-d discrete Schr\"odinger operators with positive weights. The distinctive feature of this work lies in the degeneracy of the weights, with both the potentials and weights assumed to be analytic and quasi-periodic. Operators of this kind originate from distinct mathematical physics problems, which include the Frenkel-Kontorova model with impurities, the discretization of singular Sturm-Liouville operators, and the Fisher-KPP lattice equation in heterogeneous media.