Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise

TL;DR

This paper constructs the integrated density of states for the 2D Anderson Hamiltonian with white noise, analyzes its asymptotics, and applies findings to the parabolic Anderson model, advancing understanding of spectral properties in random environments.

Contribution

It provides a rigorous construction of the integrated density of states for the 2D Anderson Hamiltonian with white noise and explores its asymptotic behavior and applications.

Findings

Convergence of eigenvalue counting measures established.

Logarithmic asymptotics of the left tail derived.

Application to moment explosion in the parabolic Anderson model.

Abstract

We construct the integrated density of states of the Anderson Hamiltonian with two-dimensional white noise by proving the convergence of the Dirichlet eigenvalue counting measures associated with the Anderson Hamiltonians on the boxes. We also determine the logarithmic asymptotics of the left tail of the integrated density of states. Furthermore, we apply our result to a moment explosion of the parabolic Anderson model in the plane.