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 $[0,1]:$ \[ D'(x)+\left[\begin{array}{cc} 0 & σ_1(x)\\ σ_2(x) & 0 \end{array} \right] D(x)=iμ\left[\begin{array}{cc} 1 & 0\\ 0 &-1 \end{array} \right]D(x),\quad D(0)=\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right], \] 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 $|\mathrm{Im}\mu |\le d$. As an application, we obtain new, sharp asymptotic formulas for eigenfunctions of Sturm-Liouville operators with singular potentials.