Uniform asymptotic expansions of solutions of second-order differential equations with two turning points and a complex spectral parameter. (English. Russian original) Zbl 0668.34057

Yakushina, I. N.

Differ. Equations 23, No. 6, 684-689 (1987); translation from Differ. Uravn. 23, No. 6, 1014-1020 (1987).

Consider the second-order differential equation with two turning points and a complex spectral parameter: \((1)\quad y''(x)+p^ 2[-q(x)- \lambda]y(x)=0\) where p is a large parameter. V. S. Buldyrev and S. Yu. Slavyanov [Vestn. Leningr. Univ. 23, 70-84 (1968)] introduced asymptotic expansions of solutions of (1) and its transformed forms. In this paper the author justifies these expansions. They are used for complex values of a spectral parameter in many physical applications. This case is here investigated.

Reviewer: J.H.Tian

MSC:

34E05	Asymptotic expansions of solutions to ordinary differential equations
34E20	Singular perturbations, turning point theory, WKB methods for ordinary differential equations

Keywords:

second-order differential equation; turning points; complex spectral parameter