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

The component functions {Ψ n ( 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, T n ( cos 2 πθ ). It permits us a particular method for the θ variable separation. Thus, component functions can be expressed as an inner product,( cos 2 πθ ) ·. A matrix block transference method is applied for the calculation of the vector. When θ is integer, {Ψ n ( ε ) is the sum of component from. 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 Ψ n can be seen as a perturbation of vectoron the sum of components from the vector. When, 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 L σ = 4(1 – |λ|), 0 < |λ| ≤ 1. In this situation, the series solution becomes critical. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)