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Abderramán Marrero, Jesús Carmelo (2007). Chebyshev expansion for the component functions of the AlmostMathieu Operator. "Proceedings in Applied Mathematics and Mechanics", v. 7 (n. 1); pp. 20400712040072. ISSN 16177061. https://doi.org/10.1002/pamm.200700870.
Title:  Chebyshev expansion for the component functions of the AlmostMathieu Operator 

Author/s: 

Item Type:  Article 
Título de Revista/Publicación:  Proceedings in Applied Mathematics and Mechanics 
Date:  December 2007 
ISSN:  16177061 
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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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 almostMathieu 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 subbands. 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.
Item ID:  2857 

DC Identifier:  https://oa.upm.es/2857/ 
OAI Identifier:  oai:oa.upm.es:2857 
DOI:  10.1002/pamm.200700870 
Official URL:  http://www3.interscience.wiley.com/journal/1179257... 
Deposited by:  Memoria Investigacion 
Deposited on:  13 Apr 2010 10:11 
Last Modified:  20 Apr 2016 12:29 