Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in $d \leq 3$

TL;DR

This paper analyzes the asymptotic behavior of the smallest eigenvalues of the continuous Anderson Hamiltonian with white noise potential in dimensions up to 3, revealing their divergence rate and linking it to the Gagliardo-Nirenberg inequality.

Contribution

It extends the understanding of eigenvalue asymptotics for the Anderson Hamiltonian to dimension 3, providing new results not previously known in this dimension.

Findings

Eigenvalues diverge to -infinity at rate (log L)^{1/(2 - d/2)}

Identifies the prefactor in terms of the Gagliardo-Nirenberg inequality constant

Provides conjectures on eigenvalue fluctuations and eigenfunction localization

Abstract

We consider the continuous Anderson Hamiltonian with white noise potential on $(-L/2,L/2)^d$ in dimension $d\le 3$, and derive the asymptotic of the smallest eigenvalues when $L$ goes to infinity. We show that these eigenvalues go to $-\infty$ at speed $(\log L)^{1/(2-d/2)}$ and identify the prefactor in terms of the optimal constant of the Gagliardo-Nirenberg inequality. This result was already known in dimensions $1$ and $2$, but appears to be new in dimension $3$. We present some conjectures on the fluctuations of the eigenvalues and on the asymptotic shape of the corresponding eigenfunctions near their localisation centers.