Schrödinger's equations in periodic potentials are usually analyzed using Bloch theorem.
Because your potential is periodic it has a Fourier series, $V = \sum V_i \, e^{inx}$ and the wave function must also have a Fourier series, $\phi(x) = \sum \psi_n(x) e^{i n x}$.
Bloch's theorem says you can write the wavefunctions as $\phi(x) = u(x) e^{i\mathbf{k} x}$ where $u(x)$ is also periodic and $\mathbf{k}$ lives in the 1st Broullin zone, the fundamental domain dual to the lattice.
The proof says that any translation of the lattice should multiply the wavefunction by a phase. This phase determines the value of $\mathbf{k}$.
One example is the Kronig-Penny potential in solid state physics:
$$ V(x) = V_0 \sum_{n \in \mathbb{Z}} \delta(x - an) $$
This potential is invariant under translations $x \mapsto x + a$ and therefore we guess $\psi(x+a) = \psi(x) e^{ika}$ for the wavefunction. Then we get a relationship between the lattice momentum and the energy:
$$ \cos \lambda = \frac{v}{2\beta} \sin \eta + \cos \beta $$
with $\lambda = ka$ and $\beta = a \sqrt{\tfrac{2mE}{\hbar}}$ and $V = \tfrac{2mV_0 a}{\hbar^2}$ (these are just formulas taken from the link).
The general framework for this type of equation is Floquet-theory which deals with equations of the type $\dot{\psi} = A(x) \psi$ with $A(x)$ periodic in $x$.
There may be integrable systems which exhibit this type of monodromy behavior.