Quasi-periodic potential and Bloch's theorem Let's look at a physical system of a particle in a one dimensional periodic potential $V(x)$. When the potential satisfies the periodicity condition of the form
$$ V(x + n b) = V(x),$$
this leads to the usual nice structure, due to the (discrete)
translational symmetry of the Hamiltonian. Using a bit of group theory
we end up with Bloch's wave functions and reduced Schrodinger equation
for the periodic part.
I am interested in the generalization of this. Let me assume I have
a 'quasi-periodic' potential of the form
$$ V(\phi(x)) = V(x), $$
where $\phi(x)$ can be some more general transformation, like e.g. non-constant translations (for each step $n$ the period $b(n)$ changes), or if there are gaps in the periodicity (e.g. two superposed periodicities).
My general question: what is known for such potentials?
To be a bit more concrete, for non-constant translations
I could have something like
$$ \phi(x) = \frac{b(0)}{b(n)} \left(x + \sum_{i=1}^n b(i)\right), $$
i.e. the period size of my lattice could change, but if I translate and rescale the lattice the potential $V(x)$ stays the same.
Given that I don't know much of solid-state physics, I would have assumed people have developed quite sophisticated tools to explore such cases. Group theory, self-similarity, renormalization and such things come to mind. Is that so? Any comments, discussion of what is done and known, or even hints in the right direction would be quite helpful.
 A: I'll address your question as two separate parts.
First part: The term 'Quasi-periodicity' would imply that a function is close to, but not exactly, periodic. However the type of function you're describing, generally, could be very far from periodic. As an example, take the domain $X$ to be a discrete lattice, then take $\phi$ to be some arbitrary permutation of the lattice. Then define $V(x) = V(y)$ for all $x$ and $y$ that are in the same cycle of the permutation. This allows functions that you would likely not consider to be close to periodic. For an even more extreme example, take $\phi(x)=x$, then any function $V$ satisfies the equation.
Second part: For your concrete example, I assume that $n$ and $b(i)$ are given. Then let
$$
\sum_{i=1}^n b(i) = C
$$
and
$$
\frac{b(0)}{b(n)} = D
$$
Thus the equation reduces to:
$$
\phi(x) = D\left(x + C\right)
$$
or just an affine transform of $x$. I'll assume $D>0$. For $D=1$ this is just the normal periodic case with period $C$. For $D\ne1$, you get the fixed point
$$
x_0 = -\frac{DC}{D-1}
$$
Which means that in the limit of recursion, $\phi(x) = \phi(x_0) = x_0$ thus $V(x) = V(x_0)$ for all $x$, i.e. your potential must be constant.
In the literature there are a number of ways that quasi-periodic potentials are defined, for example cases where there are small perturbations to the periodic potential, or where there are two competing periodic potentials. No general solution is known for many of these cases and even numerically calculated spectra can be very intricate. See the references in this article for many different examples.
