The functional structure of the monodromy matrix for Harper’s equation

Abstract

In this paper we continue our investigation of Harper’s equation:

$$ \frac{{\psi (x + h) + \psi (x - h)}}{2} + \cos \,x\;\psi (x) = E\psi (x). $$

((1.1))

Here h is a fixed positive parameter and x ∈ ℝ or x ∈ ℂ. This equation appeared as a model for Bloch electron in a weak constant magnetic field [Ho]. The structure of the spectrum σ _h of Harper’s equation on L ₂(ℝ) appeared to be very rich and Harper’s equation attracted the attention of both physicists and mathematicians, see, for example, [C-F-K-S].

The work was supported by the Russian Foundation of Fundamental Research