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

We consider the Dirac system of ordinary differential equations $$Y^{\mathrm{\prime }}(x)+\left[\begin{array}{cc}0& {\sigma }_1(x)\\ {\sigma }_2(x)& 0\end{array}\right]Y(x)=i\mu \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]Y(x),Y(x)=\left[\begin{array}{c}{y}_1(x)\\ y_2(x)\end{array}\right],$$ 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 $p\in [1,2)\mathrm{.}$ We study the asymptotic behavior of the system’s fundamental solutions as $\mathrm{\mid }\mu \mathrm{\mid }\to \mathrm{\infty }$ in the half-plane $\mathrm{Im}\mu >-r,$ where $r\ge 0$ 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.