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 $[0,1]:$ $D'(x)+\left[\begin{array}{cc} 0 & σ_1(x)\\ σ_2(x) & 0 \end{array} \right] D(x)=iμ\left[\begin{array}{cc} 1 & 0\\ 0 &-1 \end{array} \right]D(x),\quad D(0)=\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right],$ where $μ\in\mathbb{C}$ is a spectral parameter, and $σ_j\in L_2[0,1]$, $j=1,2$. We propose a new approach for the study of asymptotic behaviour of its solutions as $μ\to \infty$ and $|{\rm Im}\,μ|\le d$. As an application, we obtain new, sharp asymptotic formulas for eigenfunctions of Sturm-Liouville operators with singular potentials.

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