Localization of the continuous Anderson Hamiltonian in $1$-d

TL;DR

This paper investigates the spectral properties of the 1D Anderson Hamiltonian with white noise, revealing Poisson eigenvalue statistics, eigenfunction localization, and universal shape, with results applicable to both Dirichlet and Neumann boundary conditions.

Contribution

It provides a detailed analysis of eigenvalue and eigenfunction behavior for the 1D Anderson Hamiltonian, including convergence to Poisson processes and explicit eigenfunction shape descriptions.

Findings

Eigenvalues form a Poisson point process with exponential intensity.

Eigenfunctions localize around random points with hyperbolic cosine shape.

Eigenvalues and eigenfunctions are similar for Dirichlet and Neumann cases.

Abstract

We study the bottom of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_x^2 + \xi$ on $[0,L]$ driven by a white noise $\xi$ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as $L\rightarrow\infty$, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on $\mathbb{R}$ with intensity $e^x dx$, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.