Spectral theory of extended Harper's model and a question by Erdos and Szekeres

• 成果类型: Article
• 署名作者: Avila, A.; Jitomirskaya, S.; Marx, C. A.
• 署名单位: Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Cite; Instituto Nacional de Matematica Pura e Aplicada (IMPA); University of California System; University of California Irvine; University System of Ohio; Oberlin College
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0729-1
• 发表日期: 2017
• 页码: 283-339
• 关键词: singular continuous-spectrum; ABSOLUTELY CONTINUOUS-SPECTRUM; rotation number; square-lattice; phase-diagram; localization; Operators; Duality

摘要

The extended Harper's model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry-Andre duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such spectral collapse,however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erdos and Szekeres (On the product Pi(k)(n) = 1(1-z(ak)), Publ. de l'Institut mathematique, Paris, 1950).