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

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.

To appear in Stochastics and Partial Differential Equations: Analysis and Computations

References in corpus (2)

• The delocalized phase of the Anderson Hamiltonian in $1$-d
• Localization crossover for the continuous Anderson Hamiltonian in $1$-d