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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.12147 |
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| _version_ | 1866914091134615552 |
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| author | Gomilko, Alexander Rzepnicki, Łukasz |
| author_facet | Gomilko, Alexander Rzepnicki, Łukasz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter Gomilko, Alexander Rzepnicki, Łukasz Functional Analysis 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. |
| title | Asymptotic behavior of solutions to the Dirac system with respect to a spectral parameter |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.12147 |