Localization crossover for the continuous Anderson Hamiltonian in $1$-d - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2021

Localization crossover for the continuous Anderson Hamiltonian in $1$-d

Résumé

We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian $\mathcal{H}_L$, with white noise potential, on a segment whose size $L$ is sent to infinity. We zoom around energy levels $E$ either of order $1$ (Bulk regime) or of order $1\ll E \ll L$ (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of $\mathcal{H}_L$. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.

Dates et versions

hal-03436108 , version 1 (19-11-2021)

Identifiants

Citer

Laure Dumaz, Cyril Labbé. Localization crossover for the continuous Anderson Hamiltonian in $1$-d. 2021. ⟨hal-03436108⟩
19 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More