On the spatial extent of localized eigenfunctions for random Schr\"odinger operators

TL;DR

This paper establishes new bounds on eigenfunctions of random Schrödinger operators, showing they are exponentially localized within finite regions with high probability, and most eigenfunctions have a bounded localization length.

Contribution

It introduces improved bounds on eigenfunction localization, defining a 'localization onset length' and quantifying the number of eigenfunctions exceeding this length.

Findings

Eigenfunctions decay exponentially outside a certain radius.

Most eigenfunctions are localized within finite regions regardless of system size.

Number of eigenfunctions with large localization length decreases exponentially.

Abstract

The present paper is devoted to new, improved bounds for the eigenfunctions of random operators in the localized regime. We prove that, in the localized regime with good probability, each eigenfunction is exponentially decaying outside a ball of a certain radius, which we call the "localization onset length". For $\ell>0$ large, we count the number of eigenfunctions having onset length larger than $\ell$ and find it to be smaller than $\exp(-C\ell)$ times the total number of eigenfunctions in the system. Thus, most eigenfunctions localize on finite size balls independent of the system size.