THE DELOCALIZED PHASE OF THE ANDERSON HAMILTONIAN IN 1-D

• 成果类型: Article
• 署名作者: Dumaz, Laure; Labbe, Cyril
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite; Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1591
• 发表日期: 2023
• 页码: 805-839
• 关键词: schrodinger-operators

摘要

We introduce a random differential operator that we call CS tau operator, whose spectrum is given by the Sch(tau) point process introduced by Kritchevski, Valko and Virag (Comm. Math Phys. (2012) 314 775-806) and whose eigen-vectors match with the description provided by Rifkind and Virag (Geom. Funct. Anal. (2018) 28 1394-1419). This operator acts on R-2-valued func-tions from the interval [0, 1] and takes the form [GRAPHICS] . where dB, dW(1) and dW(2) are independent white noises. Then we investigate the high part of the spectrum of the Anderson Hamiltonian H-L:= -partial derivative(2)(t) + d B on the segment [0, L] with white noise potential dB, when L -> infinity. We show that the operator H-L, recentred around energy levels E similar to L/tau and unitarily transformed, converges in law as L -> infinity to CS tau in an appropriate sense. This allows us to answer a conjecture of Rifkind and Virag on the behavior of the eigenvectors of H-L. Our approach also explains how such an operator arises in the limit of H-L. Finally we show that, at higher energy levels, the Anderson Hamiltonian matches (asymptotically in L) with the unperturbed Laplacian -partial derivative(2)(t). In a companion paper, it is shown that, at energy levels much smaller than L, the spectrum is localized with Poisson statistics: the present paper, identifies the delocalized of the Anderson Hamiltonian.