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

The paper is concerned with the asymptotic expansion of solutions to the following 2 × 2 Dirac type system: \(\begin{array}{ccccc}Ly=-{iB}^{-1}{y}^{\mathrm{^{\prime}}}+Q\left(x\right)y=\lambda y,& B=\left(\begin{array}{cc}{b}_{1}& 0\\ 0& {b}_{2}\end{array}\right),& y=\mathrm{col}\left({y}_{1},{y}_{2}\right),& & \left(0.1\right)\end{array}\) $$\begin{array}{ccccc}Ly=-{\mathit{iB}}^{-1}y_{}^{_{}^{\prime }}+Q\left(x\right)y=\lambda y,& B=\left(\begin{array}{cc}{b}_1& 0\\ 0& b_2\end{array}\right),& y=\mathrm{col}\left(y_1,,,y_2\right),& & \left(0.1\right)\end{array}$$

with a smooth matrix potential \(Q\in {W}_{1}^{n}\left[\mathrm{0,1}\right]\otimes {\mathbb{C}}^{2\times 2}\) $Q\in W_1^n\left[0,1\right]\otimes {\mathbb{C}}^{2\times 2}$ and b₁ < 0 < b₂. If b₂ = −b₁ = 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.