I am currently working on a form of the discrete Schrödinger equation, the Harper equation:
$\psi_{m+1}-\psi_{m-1}+2\cos(2\pi \alpha m +\theta) \psi_{m}=E\psi_{m}$,
where $ m\in \mathbb{Z},\quad \alpha,\theta\in \mathbb{R}.$ $E$ is the (Energy)eigenvalue. So, this is a second order difference equation.
Now, I need to prove, that this equation can be written in a $q$-difference equation in the following form:
$ i(z^{-1}+qz)\Psi(qz)-i(zq^{-1}+z^{-1})\Psi(q^{-1}z)=E\Psi(z)$,
where $\Psi(z)=\sum_{n=0}^{Q-1}p_{n}z^{n}$ is a polynomial.
There is a paper http://arxiv.org/abs/cond-mat/9808066v2, where they do explain the procedure a bit. But I cannot really follow. I know, that this difference equation can be written as an eigenvalue problem. But I do not understand the further explanations/ derivation.
Does anybody have ideas or knows how ich can solve the problem?
Thanks a lot in advance.