Chebyshev expansion for the component functions of the almost‐Mathieu operator

Abstract

AbstractThe component functions {Ψn (ε)} (n 2 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 (cos 2πθ). It permits us a particular method for the θ variable separation. Thus, component functions can be expressed as an inner product, equation image (cos 2πθ) · equation image . A matrix block transference method is applied for the calculation of the vector equation image . When θ is integer, {Ψn (ε) is the sum of component from equation image . 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 equation image on the sum of components from the vector equation image . When equation image , 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. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)