On the Asymptotic Expansion of the Characteristic Determinant for a 2 × 2 Dirac Type System

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 284; no. 6; pp. 795 - 823
• Main Authors: Lunev, A., Malamud, M.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2024; Springer Nature B.V
• Subjects: Asymptotic series; Boundary conditions; Boundary value problems; Dirac equation; Mathematics; Mathematics and Statistics
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-024-07390-9

The paper is concerned with the asymptotic expansion of solutions to the following 2 × 2 Dirac type system: L y = - iB - 1 y ′ + Q x y = λ y , B = b 1 0 0 b 2 , y = col y 1 , y 2 , 0.1 with a smooth matrix potential Q ∈ W 1 n 0 , 1 ⊗ C 2 × 2 and b 1 < 0 < b 2 . If b 2 = − b 1 = 1, this equation is equivalent to one dimensional Dirac equation. We apply these formulas to get the asymptotic expansion of the characteristic determinant of the boundary value problem associated with the above equation subject to the general two-point boundary conditions. This expansion directly yields new completeness result for the system of root functions of such BVP with nonregular boundary conditions.