A crystal with a singular potential in a homogeneous electric field

We study the asymptotic behavior of solutions to the one-dimensional Schrödinger equation $-\psi''+q(x)\psi-Fx\psi=E\psi$ for large arguments. We assume that the potential $q$ is a periodic function and is absolutely integrable over the period. We show that the spectral problem for the original Schrödinger equation can be reduced to the spectral problem for a discrete system. If the potential $q$ is smooth, the transition matrix tends to the unit matrix rapidly; if $q$ is not smooth, the transition matrix tends to the unit matrix slowly, and the discrete system demonstrates random properties. This explains why the spectrum of the original equation has remained practically unexplored.