Localization crossover for the continuous Anderson Hamiltonian in 1-d
Dumaz, Laure; Labbé, Cyril (2021), Localization crossover for the continuous Anderson Hamiltonian in 1-d. https://basepub.dauphine.psl.eu/handle/123456789/22792
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-03436108
Series titleCahier de recherche CEREMADE, Université Paris Dauphine-PSL
MetadataShow full item record
Département de Mathématiques et Applications - ENS Paris [DMA]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian HL, 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≪E≪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 HL. 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.
Subjects / KeywordsAnderson Hamiltonian; Hill’s operator; localization; diffusion; Poisson statistics; hypocoercivity; Malliavin calculus
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