Poisson statistics for 1d Schr\"odinger operators with random decaying potentials

TL;DR

This paper demonstrates that for 1D Schrödinger operators with decaying random potentials, the rescaled eigenvalues and eigenfunction zeros form a Poisson process, revealing a fundamental statistical property of the spectrum.

Contribution

It establishes the convergence of the combined eigenvalues and eigenfunction zeros to a Poisson process in the sub-critical case with decaying potentials.

Findings

Eigenvalues and eigenfunction zeros form a Poisson process

Spectrum is pure point in the sub-critical case

Rescaled eigenvalues and zeros exhibit Poisson statistics

Abstract

We consider the 1d Schr\"odinger operators with random decaying potentials where the spectrum is pure point(sub-critical case). We show that the point process composed of the rescaled eivenvalues, together with those zero points of the corresponding eigenfunctions, converges to the Poisson process.