Adiabatic perturbation of a periodic potential

A differential equation of the form $\left[-\frac{d^2}{dx^2}+p(x)+q(\varepsilon x)\right]f=0$ is considered. The coefficient $p$ is assumed to be a periodic function: $p(x+a) =p(x)$. The behavior of the solutions for $|\varepsilon|\ll1$ is studied. The concept of a turning point is generalized to this case, and self-consistent asymptotic expressions are obtained for the solutions at a certain distance from the turning points and in their neighborhoods. For $p=0$, the obtained expressions agree with the classical WKB expressions.