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Chebyshev expansion for the component functions of the Almost-Mathieu Operator

Authors
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E.T.S.I. Telecomunicación (UPM)
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Keywords
  • Matemáticas

Abstract

The component functions {Ψn(∈)} (n ∈ Z+) from difference Schrödinger operators, can be formulated in a second order linear difference equation. Then the Harper equation, associated to almost-Mathieu operator, is a prototypical example. Its spectral behavior is amazing. Here, due the cosine coefficient in Harper equation, the component functions are expanded in a Chebyshev series of first kind, Tn(cos2πθ). It permits us a particular method for the θ variable separation. Thus, component functions can be expressed as an inner product, Ψn(, λ, θ) = _T [ n(n−1) 2 ] (cos2πθ) • _A [ n(n−1) 2 ] (_, λ). A matrix block transference method is applied for the calculation of the vector _A [ n(n−1) 2 ] (_, λ). When θ is integer, Ψn(_) is the sum of component from _A [ n(n−1) 2 ]. The complete family of Chebyshev Polynomials can be generated, with fit initial conditions. The continuous spectrum is one band with Lebesgue measure equal to 4. When θ is not integer the inner product Ψn can be seen as a perturbation of vector _T [ n(n−1) 2 ] on the sum of components from the vector _A [ n(n−1) 2 ]. When θ = p q , with p and q coprime, periodic perturbation appears, the connected band from the integer case degenerates in q sub-bands. When θ is irrational, ergodic perturbation produces that one band spectrum from integer case degenerates to a Cantor set. Lebesgue measure is Lσ = 4(1 − |λ|), 0 < |λ| ≤ 1. In this situation, the series solution becomes critical.

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