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Article Dans Une Revue Stochastics and Partial Differential Equations: Analysis and Computations Année : 2022

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

Résumé

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.

Dates et versions

hal-03427135 , version 1 (12-11-2021)

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Yueh-Sheng Hsu, Cyril Labbé. Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in $d \leq 3$. Stochastics and Partial Differential Equations: Analysis and Computations, 2022, ⟨10.1007/s40072-022-00252-y⟩. ⟨hal-03427135⟩
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