On the 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]:

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.