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

Abstract: The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval $[0,1]:$ [ D'(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),\quad 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 $\mu\to \infty$ and $|{\rm Im}\,\mu|\le d$. As an application, we obtain new, sharp asymptotic formulas for eigenfunctions of Sturm-Liouville operators with singular potentials.