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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.

MSC:

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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