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 $\mathbf{k}$.