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

TL;DR

This paper develops a new method to analyze the asymptotic behavior of solutions to the Dirac system with singular potentials, leading to improved asymptotic formulas for Sturm-Liouville eigenfunctions with such potentials.

Contribution

It introduces a novel approach for studying Dirac system solutions asymptotics as spectral parameter grows large, with applications to Sturm-Liouville problems with singular potentials.

Findings

New asymptotic formulas for Sturm-Liouville eigenfunctions

Effective method for Dirac system asymptotics as spectral parameter tends to infinity

Enhanced understanding of solutions with singular potentials

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.