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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:
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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