Hofstadter point spectrum trace and the almost Mathieu operator
نویسندگان
چکیده
منابع مشابه
The Spectrum of the Almost Mathieu Operator
The notes are based on a series of six lectures, given during my stay at the CRC 701 in June/July 2008. The lecture series intended to give a survey of some of the results for the almost Mathieu operator that have been obtained since the early 1980’s. Specifically, the metalinsulator transition is discussed in detail, along with its relation to the ten Martini problem via duality and reducibility.
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We review some aspects of the spectral theory of the Almost Mathieu operator (acting on`2 (Z)): where ; ; 2 R. We concentrate on the spectrum as a set, and describe the partial results obtained so far on the conjectures that, for irrational , it is a Cantor set and has Lebesgue measure j4 ? 2jjj.
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Recently, there has been an explosion of interest in the study of Schrijdinger operators and Jacobi matrices with almost periodic potential (see, e.g., the review [ 161). The general belief is that generically the spectrum is a Cantor set, i.e., a nowhere dense perfect, closed set. Since it is easy to prove that the spectrum is closed and perfect (see, e.g., [2]), the key is to prove that the s...
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In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx)n = xn+1 + xn−1 + b cos (2πnω + φ) xn on l2(Z) and its associated eigenvalue equation to deduce that for b 6= 0,±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called “Ten Martini Problem” for these values of b and ω. Moreover, ...
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ژورنال
عنوان ژورنال: Journal of Mathematical Physics
سال: 2018
ISSN: 0022-2488,1089-7658
DOI: 10.1063/1.5020147