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Memorias de investigación
Research Publications in journals:
Chebyshev expansion for the component functions of the Almost-Mathieu Operator
Year:2008
Research Areas
  • Mathematics
Information
Abstract
The component functions {Ψn(e)} (n ∈ Z+) from difference Schr¨odinger 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(e, λ, θ) = eT [ n(n−1) 2 ] (cos2πθ) · A [ n(n−1) 2 ] (e, λ). A matrix block transference method is applied for the calculation of the vector A [ n(n−1) 2 ] (e, λ). When θ is integer, Ψn(e) 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 = pq , 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.
International
Si
JCR
No
Title
PAMM Proc. Appl. Math. Mech. (Wiley)
ISBN
1617-7061
Impact factor JCR
0
Impact info
Volume
7
Journal number
0
From page
2040071
To page
2040072
Month
DICIEMBRE
Ranking
Participants
  • Autor: Jesus Carmelo Abderraman Marrero (UPM)
Research Group, Departaments and Institutes related
  • Creador: Grupo de Investigación: Polinomios Ortogonales y Geometría Fractal
  • Departamento: Matemática Aplicada (Facultad de Informática)
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