How the Angle of Static Field Affects the Spectrum of Lattice Schrodinger Operator on ${\mathbb{Z}}^2$

Abstract

The spectral properties of the discrete Schrodinger operator on a two-dimensional lattice under uniform static fields are critically governed by the field direction: for angles parameterized by irrational frequencies, the operator is exponentially localized, and for a subset of such angles with asymptotically full Lebesgue measure, this localization persists under a given logarithmically decaying perturbation potential.

29 pages