Geodesic
Adiabatic perturbation of a periodic potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 58 (1984) no. 2, pp. 233-243

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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. 
@article{TMF_1984_58_2_a8,
     author = {V. S. Buslaev},
     title = {Adiabatic perturbation of a periodic potential},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {233--243},
     publisher = {mathdoc},
     volume = {58},
     number = {2},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1984_58_2_a8/}
}
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V. S. Buslaev. Adiabatic perturbation of a periodic potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 58 (1984) no. 2, pp. 233-243. http://geodesic.mathdoc.fr/item/TMF_1984_58_2_a8/