Spectral asymptotics for solutions of $2\times 2$ system of ordinary differential equations of the first order

The aim of the paper is to find representation for solutions of $2\times 2$ system of ordinary differential equations $${\mathbf{y}}^{\mathbf{\prime }}-B(x)\mathbf{y}=\lambda A(x)\mathbf{y},x\in [0,1],$$ where $A(x)=diag\{a_1(x),a_2(x)\}$, $B(x)=(b_{ij}(x))$, $a_1(x)>0,a_2(x)<0$ and all the functions $a_i,b_{ij}$ belong to the Sobolev spaces $W_1^n[0,1]$ for given integer $n⩾0$. We prove that there exists a fundamental matrix of solutions for the above system, which have representation $$Y(x,\lambda )=M(x)(I+\frac{R^1(x)}{\lambda }+\cdots +\frac{R^n(x)}{{\lambda }^n}+o(1){\lambda }^{-n})E(x,\lambda ),$$ where $o(1)\to 0$ uniformly for $x\in [0,1]$ as the spectral parameter $\lambda \to \mathrm{\infty }$ in the half plane $\mathrm{\Re }\lambda >-\kappa$ or $\mathrm{\Re }\lambda <\kappa$, where $\kappa$ is any fixed real number. The main novelty is that we give explicit formulae for all matrices $M,E$ and $R^m$ in this representation.