Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter

TL;DR

This paper investigates the asymptotic behavior of solutions to a Dirac system with spectral parameter, providing detailed formulas and applications to Sturm--Liouville equations with singular potentials.

Contribution

It offers new asymptotic formulas for Dirac system solutions as the spectral parameter grows large, extending understanding to systems with less regular potentials.

Findings

Derived detailed asymptotic formulas for fundamental solutions

Extended asymptotic analysis to Sturm--Liouville equations with singular potentials

Provided insights into the behavior of solutions in the complex spectral plane

Abstract

We consider the Dirac system of ordinary differential equations \[ Y'(x) + \begin{bmatrix} 0 & \sigma_1(x) \\ \sigma_2(x) & 0 \end{bmatrix} Y(x) = i\mu \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} Y(x), \quad Y(x) = \begin{bmatrix} y_1(x) \\ y_2(x) \end{bmatrix}, \] where $x \in [0,1]$, $\mu \in \mathbb{C}$ is a spectral parameter, and $\sigma_j \in L^p[0,1],$ $j = 1,2,$ for $p \in [1,2).$ We study the asymptotic behavior of the system's fundamental solutions as $|\mu| \to \infty$ in the half-plane $\operatorname{Im} \mu > -r,$ where $r \geq 0$ is fixed, and obtain detailed asymptotic formulas. As an application, we derive new results on the half-plane asymptotics of fundamental solutions to Sturm--Liouville equations with singular potentials.