Chebyshev expansion for the component functions of 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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Let Aθ be the rotation C*-algebra for angle θ. For θ = p/q with p and q relatively prime, Aθ is the sub-C*-algebra of Mq(C(T )) generated by a pair of unitaries u and v satisfying uv = e2πiθvu. Let hθ,λ = u+u +λ/2(v+ v) be the almost Mathieu operator. By proving an identity of rational functions we show that for q even, the constant term in the characteristic polynomial of hθ,λ is (−1) q/2(1 + ...
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We prove that for Diophantine ω and almost every θ, the almost Mathieu operator, (Hω,λ,θΨ)(n) = Ψ(n+ 1) + Ψ(n− 1) +λ cos 2π(ωn+ θ)Ψ(n), exhibits localization for λ > 2 and purely absolutely continuous spectrum for λ < 2. This completes the proof of (a correct version of) the Aubry-André conjecture.
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ژورنال
عنوان ژورنال: PAMM
سال: 2007
ISSN: 1617-7061,1617-7061
DOI: 10.1002/pamm.200700870