Adiabatic asymptotics of the reflection coefficient. (English. Russian original) Zbl 1084.34026

St. Petersbg. Math. J. 16, No. 3, 437-452 (2005); translation from Algebra Anal. 16, No. 3, 1-23 (2004).
Considered is the Sturm-Liouville problem \[ -\psi''+p(x,\xi)\psi=E\psi,\;x\geq0, \;\psi(0)=0, \] with a parameter \(\xi\geq0\). The behaviour of the solutions as \(\xi\to\infty \) and \(\varepsilon \to0\) in \(\xi=\varepsilon x\) is investigated. The various branches of the band function \(\mathcal E(k,\xi)\) play a crucial role in this paper, and particular attention is given to turning points. Formal solutions through asymptotic expansions are shown to be true solutions. Finally, the asymptotics of the reflection coefficient is studied.


34B24 Sturm-Liouville theory
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
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[1] Спектрал\(^{\приме}\)ная теория операторов Штурма-Лиувилля., Издат. ”Наукова Думка”, Киев, 1972 (Руссиан).
[2] Линейные дифференциал\(^{\приме}\)ные уравнения с периодическими коѐффициентами и их приложения, Издат. ”Наука”, Мосцощ, 1972 (Руссиан). В. А. Якубович анд В. М. Старжинскии, Линеар дифферентиал ечуатионс щитх периодиц цоеффициенц. 1, 2, Халстед Пресс [Јохн Щилеы & Сонс] Нещ Ыорк-Торонто, Онт.,; Исраел Програм фор Сциентифиц Транслатионс, Јерусалем-Лондон, 1975. Транслатед фром Руссиан бы Д. Лоувиш.
[3] V. S. Buslaev, Adiabatic perturbation of a periodic potential, Teoret. Mat. Fiz. 58 (1984), no. 2, 233 – 243 (Russian, with English summary). · Zbl 0534.34064
[4] V. S. Buslaev and L. A. Dmitrieva, A Bloch electron in an external field, Algebra i Analiz 1 (1989), no. 2, 1 – 29 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 2, 287 – 320. · Zbl 0714.34128
[5] Vladimir Buslaev and Alain Grigis, Imaginary parts of Stark-Wannier resonances, J. Math. Phys. 39 (1998), no. 5, 2520 – 2550. · Zbl 1001.34075
[6] Vladimir Buslaev and Alain Grigis, Turning points for adiabatically perturbed periodic equations, J. Anal. Math. 84 (2001), 67 – 143. · Zbl 0987.35013
[7] V. S. Buslaev, Quasiclassical approximation for equations with periodic coefficients, Uspekhi Mat. Nauk 42 (1987), no. 6(258), 77 – 98, 248 (Russian).
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