Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter
Alexander Gomilko, Łukasz Rzepnicki
TL;DR
The paper investigates sharp asymptotics for the fundamental solutions of a one-dimensional Dirac system with integrable potentials as the spectral parameter $\mu$ grows large in the half-plane $\Sigma_r=\{\mu\in\mathbb{C}:\operatorname{Im}\mu>-r\}$. It develops an integral-equation framework featuring μ-analytic operators $K_1(\mu)$ and $K_2(\mu)$, and derives three-tier asymptotics for the fundamental system $W,V$ with precise remainder controls $\Xi_q$ and $\Lambda_q$, valid uniformly in $x$ and in $\mu$ within the half-plane. The analysis extends to the Cauchy problem and to perturbed Dirac systems, yielding refined representations with $O(|\mu|^{-1})$ corrections and stable boundary-condition behavior. An important application relates these Dirac-system asymptotics to Sturm--Liouville problems with singular potentials via a regularization that produces accurate high-energy expansions for the Sturm--Liouville fundamental solutions and quasi-derivatives. Overall, the results enhance non-self-adjoint spectral theory for Dirac operators and provide sharp tools for studying basis properties and spectral data in problems with distributional potentials.
Abstract
We consider the Dirac system of ordinary differential equations \[ Y'(x) + \begin{bmatrix} 0 & σ_1(x) \\ σ_2(x) & 0 \end{bmatrix} Y(x) = iμ\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} Y(x), \quad Y(x) = \begin{bmatrix} y_1(x) \\ y_2(x) \end{bmatrix}, \] where $x \in [0,1]$, $μ\in \mathbb{C}$ is a spectral parameter, and $σ_j \in L^p[0,1],$ $j = 1,2,$ for $p \in [1,2).$ We study the asymptotic behavior of the system's fundamental solutions as $|μ| \to \infty$ in the half-plane $\operatorname{Im} μ> -r,$ 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.