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

TL;DR

This paper derives explicit spectral asymptotic formulas for solutions of a first-order 2x2 differential system with variable coefficients, providing detailed representations and uniform estimates as the spectral parameter grows large.

Contribution

It introduces explicit formulas for the matrices in the asymptotic solution representation of a 2x2 differential system, extending previous results with uniform estimates.

Findings

Existence of a fundamental matrix with asymptotic expansion

Explicit formulas for matrices in the solution representation

Uniform asymptotic estimates as spectral parameter tends to infinity

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.