Asymptotic behavior of solutions of the Dirac system with an integrable potential

TL;DR

This paper investigates the asymptotic behavior of solutions to a Dirac system with an integrable potential, deriving precise formulas for eigenvalues and eigenfunctions of related Sturm–Liouville operators.

Contribution

It provides new asymptotic formulas for solutions, eigenvalues, and eigenfunctions of Dirac and Sturm–Liouville systems with $L_p$ potentials for $1 \\leq p < 2$.

Findings

Sharp asymptotic formulas for eigenvalues.

Asymptotic behavior of solutions in a complex stripe.

Connections between Dirac and Sturm–Liouville operators.

Abstract

We consider the Dirac system on the interval $[0,1]$ with a spectral parameter $\mu\in\mathbb{C}$ and a complex-valued potential with entries from $L_p[0,1]$, where $1\leq p <2$. We study the asymptotic behavior of its solutions in a stripe $|{\rm Im}\,\mu|\le d$ for $\mu\to \infty$. These results allows us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm--Liouville operators associated with the aforementioned Dirac system.