On an asymptotic expansion of the characteristic determinant for $2 \times 2$ Dirac type systems

The paper is concerned with the asymptotic expansion of solutions to the following $2 \times 2$ Dirac type system
$$ L y = -i B^{-1} y' + Q(x) y = \lambda y,\quad B = \begin{pmatrix} b_1 & 0\\ 0 & b_2 \end{pmatrix},\quad y= \mathrm{col}(y_1, y_2), $$
with a smooth matrix potential $Q \in W_1^n[0,1] \otimes \mathbb{C}^{2 \times 2}$ and $b_1 < 0 < b_2$. If $b_2 = -b_1 =1$, this equation is equivalent to the one-dimensional Dirac equation.
These formulas are applied to get an 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 a new completeness result for the system of root functions of such a boundary-value problem with nonregular boundary conditions.