In some cases, the Schrödinger equation can be solved analytically on a one-dimensional lattice of finite length using the theory of periodic differential equations. The length of the lattice is assumed to be L = N a , where a is the potential period and the number of periods N is a positive integer. The two ends of the lattice are at τ τ and L + τ τ , where τ τ determines the point of termination. The wavefunction vanishes outside the interval [ τ τ , L + τ τ ] .