Localization crossover for the continuous Anderson Hamiltonian in $1$-d

TL;DR

This paper studies the spectral behavior of the 1D continuous Anderson Hamiltonian with white noise potential as the system size grows, revealing Poisson eigenvalue statistics, eigenfunction localization, and phase transition insights.

Contribution

It provides a detailed analysis of eigenvalue distributions and eigenfunction limits in the bulk and crossover regimes, advancing understanding of localization-delocalization transition.

Findings

Eigenvalues form a Poisson point process in the limit

Eigenfunctions exhibit exponential localization at explicit rates

Universal limits for eigenfunctions in the crossover regime

Abstract

We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $\mathcal{H}_L$, with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order $1$ (Bulk regime) or of order $1\ll E \ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of $\mathcal{H}_L$. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.