Opuscula Math. 45, no. 3 (2025), 403-416
https://doi.org/10.7494/OpMath.2025.45.3.403

 
Opuscula Mathematica

Integral representation of solutions to Dirac systems

Łukasz Rzepnicki

Abstract. We introduce a novel integral form for a fundamental set of solutions to one-dimensional Dirac systems with an integrable potential and spectral parameter \(\mu \in \mathbb{C}\). This method enables the construction of solutions that are analytic in \(\mu\) within the half-plane \(\operatorname{Im} \mu\gt -r\), \(r\geq 0\) and \(|\mu| \to \infty\). Consequently, we derive estimates for the solutions that remain valid not just within a horizontal strip but throughout the entire half-plane.

Keywords: Dirac system, integrable potential, integral equations, fundamental system of solutions.

Mathematics Subject Classification: 34L40, 45F05.

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  • Communicated by P.A. Cojuhari.
  • Received: 2025-02-25.
  • Revised: 2025-04-29.
  • Accepted: 2025-05-02.
  • Published online: 2025-05-30.
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Cite this article as:
Łukasz Rzepnicki, Integral representation of solutions to Dirac systems, Opuscula Math. 45, no. 3 (2025), 403-416, https://doi.org/10.7494/OpMath.2025.45.3.403

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