Limit-Periodic Schrodinger Operators on Z^d: Uniform Localization

TL;DR

This paper constructs d-dimensional limit-periodic Schrödinger operators on Z^d that demonstrate uniform exponential localization and dynamical localization, with a consistent decay rate across the entire hull of the operators.

Contribution

It introduces a class of limit-periodic Schrödinger operators in multiple dimensions that are uniformly localized, a significant strengthening of localization results.

Findings

Operators exhibit uniform exponential decay of eigenfunctions.

All elements in the hull have a complete set of exponentially localized eigenvectors.

Operators demonstrate uniform dynamical localization.

Abstract

We exhibit d-dimensional limit-periodic Schrodinger operators that are uniformly localized in the strongest sense possible. That is, for each of these operators, there is a uniform exponential decay rate such that every element of the hull of the corresponding Schrodinger operator has a complete set of eigenvectors that decay exponentially off their centers of localization at least as fast as prescribed by the uniform decay rate. Consequently, these operators exhibit uniform dynamical localization.