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

2025/07/16 by A. M. Gomilko, Gomilko, Alexander, Łukasz Rzepnicki +1
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2507.12147

openalex publication_date 2025/07/16 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

We consider the Dirac system of ordinary differential equations Y'(x) + \beginbmatrix 0 amp; σ₁(x)
σ₂(x) amp; 0 \endbmatrix Y(x) = iμ\beginbmatrix 1 amp; 0
0 amp; -1 \endbmatrix Y(x), Y(x) = \beginbmatrix y₁(x)
y₂(x) \endbmatrix, where x ∈ [0,1], μ∈ ℂ is a spectral parameter, and σ_j ∈ L^p[0,1], j = 1,2, for p ∈ [1,2). We study the asymptotic behavior of the system's fundamental solutions as |μ| → ∞ in the half-plane Im μ> -r, where r ≥ 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.

Citations

• On the completeness of root function system of the Dirac operator with two-point boundary conditions
• Stability of spectral characteristics and Bari basis property of boundary value problems for 2 × 2 Dirac type systems
• On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators
• Riesz bases consisting of root functions of 1D Dirac operators
• Sturm-Liouville operators with distributional potentials

Related

• On asymptotic behaviour of solutions of the Dirac system and\n applications to the Sturm-Liouville problem with a singular potential
• Asymptotic behavior of solutions of the Dirac system with an integrable\n potential
• Spectral analysis of the Dirac system with a singularity in an interior point
• Asymptotics of eigenvalues and eigenfunctions of energy-dependent Sturm--Liouville equations
• Spectral asymptotics for solutions of 2× 2 system of ordinary differential equations of the first order