Asymptotic comparison of differential equations. (English. Russian original) Zbl 1055.34104

St. Petersbg. Math. J. 14, No. 4, 535-547 (2003); translation from Algebra Anal. 14, No. 4, 1-18 (2002).
The paper deals with the asymptotic behavior as \(\varepsilon\to 0\) of the solutions to the linear differential equation \[ i\varepsilon dx/dt= A(t)x, \] where \(A(t): \mathbb{C}^2\to \mathbb{C}^2\) is a given smooth operator valued function, \(t\in [\alpha,\beta]\subset \mathbb{R}\) and \(x(t,\varepsilon)\in \mathbb{C}^2\). Let \(k(t)\) be an eigenvalue of the operator \(A(t)\). The point in which \(k(t)= 0\) is called a turning point. The asymptotic bahaviour of the solutions to the equation is considered in two cases:
(i) there exists only one turning point in \([\alpha,\beta]\), (ii) there are two turning points in \([\alpha,\beta]\).
The author gives simple asymptotic formulas in these cases.


34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34A30 Linear ordinary differential equations and systems
34E10 Perturbations, asymptotics of solutions to ordinary differential equations