Uniform localization is always uniform

TL;DR

This paper proves that if a family of ergodic Schrödinger operators with continuous potentials has uniformly localized eigenfunctions, then these eigenfunctions are necessarily uniformly localized in a homogeneous way.

Contribution

It establishes a fundamental link between uniform localization and homogeneous localization for ergodic Schrödinger operators.

Findings

Uniform localization implies homogeneous localization.

Eigenfunctions are uniformly localized in a homogeneous sense.

The result applies to ergodic Schrödinger operators with continuous potentials.

Abstract

In this note we show that if a family of ergodic Schr\"odinger operators on $l^2({\Bbb Z}^\gamma)$ with continuous potentials have uniformly localized eigenfunctions then these eigenfunctions must be uniformly localized in a homogeneous sense.