Asymptotics in the Spectral Parameter for Solutions of $2 \times 2$ Systems of Ordinary Differential Equations

We consider a $2 \times 2$ system of ordinary differential equations
$$ y'-By=\lambda Ay, \qquad y=y(x), \quad x \in [0, 1], $$
where $A=\operatorname{diag}\{a_1(x), a_2(x)\}$, $B=\{b_{kj}(x)\}_{k, j=1}$, and all functions occurring in the matrices are complex-valued and integrable. In the case
$$ a_1,a_2, b_{21},b_{12} \in W^n_1[0,1], \qquad b_{11}, b_{22} \in W^{n-1}_1[0,1], $$
we obtain $n+1$ terms of the asymptotic expansion in powers of $\lambda^{-1}$, $\lambda \to \infty$, of the fundamental matrix of solutions of this equation. These asymptotic expansions are valid in the half-planes $\Pi_{\kappa}=\{\lambda \in \mathbb{C} \mid \operatorname{Re}{\lambda} \ge -\kappa \}$, $\kappa \in \mathbb{R}$, and $-\Pi_{\kappa}$ if $a_1(x)-a_2(x) >0$. They hold in the sectors $S=\{\lambda \in \mathbb{C} \mid \lvert\operatorname{arg}\lambda\rvert \le \pi/2-\phi-\varepsilon\}$, $\varepsilon > 0$, and $-S$ under the condition that $\lvert\operatorname{arg}\{a_1(x)-a_2(x)\}\rvert<\phi<\pi /2$. The main novelty of the work is that we assume minimal conditions for the smoothness of the functions and in addition we obtain explicit formulae for matrices involved in asymptotic expansions. The results are also new for the Dirac system.