Localization of the continuous Anderson Hamiltonian in 1-D

• 成果类型: Article
• 署名作者: Dumaz, Laure; Labbe, Cyril
• 署名单位: Universite PSL; Universite Paris-Dauphine; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-019-00920-6
• 发表日期: 2020
• 页码: 353-419
• 关键词: ground-state eigenvalue; RANDOM MATRICES; tracy-widom; spectrum; diffusion

摘要

We study the bottom of the spectrum of the Anderson Hamiltonian H-L := -partial derivative(2)(x) + xi on [0, L] driven by a white noise xi and endowed with either Dirichlet or Neumann boundary conditions. We show that, as L -> infinity, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on R with intensity e(x)dx, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentered around its maximum and properly rescaled, is given by the inverse of a hyperbolic cosine. We also show that the eigenfunctions decay exponentially from their localization centers at an explicit rate, and we obtain very precise information on the zeros and local maxima of these eigenfunctions. Finally, we show that the eigenvalues/eigenfunctions in the Dirichlet and Neumann cases are very close to each other and converge to the same limits.