On asymptotic behaviour of solutions of the Dirac system and applications to the Sturm-Liouville problem with a singular potential

The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval $[0,1]:$ $$D^{\mathrm{\prime }}(x)+\left[\begin{array}{cc}0& {\sigma }_1(x)\\ {\sigma }_2(x)& 0\end{array}\right]D(x)=i\mu \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]D(x),D(0)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],$$ where $\mu \in \mathbb{C}$ is a spectral parameter, and ${\sigma }_j\in L_2[0,1]$, $j=1,2$. We propose a new approach for the study of asymptotic behaviour of its solutions as $\mu \to \mathrm{\infty }$ and $\mathrm{\mid }\mathrm{I}\mathrm{m}\mu \mathrm{\mid }\le d$. As an application, we obtain new, sharp asymptotic formulas for eigenfunctions of Sturm-Liouville operators with singular potentials.