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

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.