Shape of eigenvectors for the decaying potential model

TL;DR

This paper analyzes the eigenvector behavior of a 1D Schrödinger operator with decaying random potential, revealing distinct limiting measures depending on the decay rate, which aligns with known spectral properties.

Contribution

It provides a comprehensive characterization of the limiting eigenfunction measures across different decay regimes, extending understanding of spectral behavior in decaying random potentials.

Findings

Lebesgue measure limit for lpha > 1/2

Power-law decay measure with Brownian fluctuation at lpha=1/2

Delta measure with uniformly distributed atom for lpha<1/2

Abstract

We consider the 1d Schr\"odinger operator with decaying random potential, and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions which is based on the formulation by Rifkind-Virag. As a result, we have completely different behavior depending on the decaying rate $\alpha > 0$ of the potential : the limiting measure is equal to (1) Lebesgue measure for the super-critical case ($\alpha > 1/2$), (2) a measure of which the density has power-law decay with Brownian fluctuation for critical case ($\alpha=1/2$), and (3) the delta measure with its atom being uniformly distributed for the sub-critical case($\alpha<1/2$). This result is consistent with previous study on spectral and statistical properties.