## On adiabatic perturbation of a periodic potential.(Russian. English summary)Zbl 0534.34064

Summary: The differential equation $-[\frac{d^ 2}{dx^ 2}+p(x)+q(\epsilon x)]f(x)=0$ is considered. It is supposed that the coefficient p is a periodic function: $$p(x+a)=p(x)$$. In the case $$| \epsilon | \ll 1$$ formulas are derived which describe the behaviour of the solutions at some distance from the turning points or in a neighbourhood of nondegenerate turning points. These formulas are natural generalisations of the well-known WKB formulas.

### MSC:

 3.4e+21 Singular perturbations, turning point theory, WKB methods for ordinary differential equations

### Keywords:

nondegenerate turning points; WKB formulas