## 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

### Keywords:

eigenvectors; turning points