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Semiclassical approximation for equations with periodic coefficients. (English. Russian original) Zbl 0698.35130

Russ. Math. Surv. 42, No. 6, 97-125 (1987); translation from Usp. Mat. Nauk 42, No. 6, 77-98 (1987).
The author gives a survey of the asymptotic theory of periodic differential (and pseudodifferential) operators with slow variables and small parameter at derivatives that are used in quantum theory of solids. A mathematical substantiation of an effective Hamiltonian notion in solid state physics is given. Turning points and appropriate problems (semiclassical quantization rules, Stark stairs, Zener breakthrough, magnetic breakdown etc.) are considered too. In the final section historical and bibliographical comments are given.
Reviewer: E.D.Belokolos

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35G99 General higher-order partial differential equations and systems of higher-order partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
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