Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter

Abstract

We consider the Dirac system of ordinary differential equations \[ Y'(x) + \begin{bmatrix} 0 & \sigma_1(x) \\ \sigma_2(x) & 0 \end{bmatrix} Y(x) = i\mu \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} Y(x), \quad Y(x) = \begin{bmatrix} y_1(x) \\ y_2(x) \end{bmatrix}, \] where $x \in [0,1]$, $\mu \in \mathbb{C}$ is a spectral parameter, and $\sigma_j \in L^p[0,1],$ $j = 1,2,$ for $p \in [1,2).$ We study the asymptotic behavior of the system's fundamental solutions as $|\mu| \to \infty$ in the half-plane $\operatorname{Im} \mu > -r,$ where $r \geq 0$ is fixed, and obtain detailed asymptotic formulas. As an application, we derive new results on the half-plane asymptotics of fundamental solutions to Sturm--Liouville equations with singular potentials.