Dynamical localization for polynomial long-range hopping random operators on $\mathbb{Z}^d$

TL;DR

This paper proves a power-law dynamical localization for a class of random operators with long-range hopping on integer lattices, showing that solutions remain localized over time in Sobolev norms.

Contribution

It establishes the first power-law dynamical localization result for polynomial long-range hopping random operators on old integer lattices.

Findings

Sobolev norm of solutions remains bounded over time

Localization persists despite long-range hopping

Extends localization theory to polynomial decay operators

Abstract

In this paper, we prove a power-law version dynamical localization for a random operator $\mathrm{H}_{\omega}$ on $\mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schr\"odinger equation $$\mathrm{i}\partial_{t}u=\mathrm{H}_{\omega}u, \quad u \in \ell^2(\mathbb{Z}^d), $$ the Sobolev norm of the solution with well localized initial state is bounded for any $t\geq 0$.