Dynamical localization and eigenvalue asymptotics: long-range hopping lattice operators with electric field

TL;DR

This paper establishes power-law dynamical localization for long-range hopping lattice operators with electric field, using novel eigenvalue asymptotics and the Min-Max Principle, without relying on traditional KAM or Green's function methods.

Contribution

It introduces new techniques for analyzing dynamical localization in long-range models with unbounded potentials, broadening applicability beyond existing methods.

Findings

Proves power-law dynamical localization for long-range hopping models.

Develops new eigenvalue asymptotics and the Power-Law ULE concept.

Applicable to models with Maryland-type potentials.

Abstract

We prove power-law dynamical localization for polynomial long-range hopping lattice operators with uniform electric field under any bounded perturbation. Actually, we introduce new arguments in the study of dynamical localization for long-range models with unbounded potentials, involving the Min-Max Principle and a notion of Power-Law ULE. Unlike existing results in the literature, our approach does not rely on KAM techniques or on Green's function estimates, but rather on the asymptotic behavior of the eigenvalues and the potential. It is worth underlining that our general results can be applied to other models, such as Maryland-type potentials.