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

Abderramán Marrero, Jesús Carmelo (2007). Chebyshev expansion for the component functions of the Almost-Mathieu Operator. "Proceedings in Applied Mathematics and Mechanics", v. 7 (n. 1); pp. 2040071-2040072. ISSN 1617-7061. https://doi.org/10.1002/pamm.200700870.

Description

Title: Chebyshev expansion for the component functions of the Almost-Mathieu Operator
Author/s:
  • Abderramán Marrero, Jesús Carmelo
Item Type: Article
Título de Revista/Publicación: Proceedings in Applied Mathematics and Mechanics
Date: December 2007
ISSN: 1617-7061
Volume: 7
Subjects:
Faculty: E.T.S.I. Telecomunicación (UPM)
Department: Matemática Aplicada a las Tecnologías de la Información [hasta 2014]
Creative Commons Licenses: Recognition - No derivative works - Non commercial

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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.

More information

Item ID: 2857
DC Identifier: http://oa.upm.es/2857/
OAI Identifier: oai:oa.upm.es:2857
DOI: 10.1002/pamm.200700870
Official URL: http://www3.interscience.wiley.com/journal/117925717/issue
Deposited by: Memoria Investigacion
Deposited on: 13 Apr 2010 10:11
Last Modified: 20 Apr 2016 12:29
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