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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.

MSC:

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
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