Perhaps it is helpful for you to know that you can find papers on $u + v + u^{\dagger} + v^{\dagger}$ also by looking for "Harper equation", "Discrete mathieu equation" or "Hofstadter butterfly".

Here's an example of the butterfly. Hoftstadter found the (rough) structure of the butterfly in 1976 by looking at a model for Bloch electrons (i.e. electrons in a periodic structure) in a magnetic field. The irrationality $\theta$ represents essentially the magnetic flux through a unit cell of the lattice.
(As for rational $p/q$ there is a translation symmetry one find $q$ "bloch bands", which touch at $E=0$ for pair $q$.)

I spent a part of my PhD thesis (no math, but renormalization from a more physical/heuristical point of view) on the multifractal properties of the spectrum for irrational values and gave some estimations on the minimal and maximal multifractal dimensions for quadratic irrationalities.
If you're interested, here's a paper of mine: http://iopscience.iop.org/0305-4470/30/1/009.