Localization crossover for the continuous Anderson Hamiltonian in 1-d

• 成果类型: Article
• 署名作者: Dumaz, Laure; Labbe, Cyril
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01225-1
• 发表日期: 2024
• 页码: 345-440
• 关键词: large disorder; SCHRODINGER-OPERATORS; spectrum; statistics; diffusion; absence

摘要

We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian 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 << 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 (Dumaz and Labbe in Ann. Probab. 51(3):805-839, 2023), this identifies completely the transition between the localized and delocalized phases of the spectrum of 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.