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

Abstract

The aim of the paper is to find representation for solutions of $2\times 2$ system of ordinary differential equations $$ \mathbf{y^\prime} - B(x)\mathbf{y} = \lambda A(x)\mathbf{y}, \quad \ x \in [0, 1], $$ 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 $W^n_1[0,1]$ for given integer $n\geqslant 0$. We prove that there exists a fundamental matrix of solutions for the above system, which have representation $$ Y(x, \lambda) = M(x)\left(I + \frac{R^1(x)}{\lambda} + \dots + \frac{R^n(x)}{\lambda^n} + o(1)\lambda^{-n}\right)E(x, \lambda), $$ where $o(1) \to 0$ uniformly for $x\in [0,1]$ as the spectral parameter $\lambda \to \infty$ in the half plane $\Re\,\lambda >-\kappa$ or $\Re\,\lambda <\kappa$, 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.