On the trace formulas and completeness property of root vectors systems for $2 \times 2$ Dirac type operators

The paper is concerned with the completeness property of the system of root vectors of a boundary value problem for the following $2 \times 2$ Dirac type equation $$L y = -i B^{-1} y' + Q(x) y = \lambda y , \quad y= {\rm col}(y_1, y_2), \quad x \in [0,1],$$ $$B = {\rm diag}(b_1, b_2), \quad b_1 < 0 < b_2, \quad\text{and}\quad Q \in W_1^n[0,1] \otimes \mathbb{C}^{2 \times 2},$$ subject to general non-regular two-point boundary conditions $C y(0) + D y(1) = 0$. If $b_2 = -b_1 = 1$ this equation is equivalent to the one dimensional Dirac equation. We establish asymptotic expansion of the characteristic determinant of this boundary value problem. This expansion directly yields new completeness result for the system of root vectors of such boundary value problem with non-regular and even degenerate boundary conditions. We also present several explicit completeness results in terms of the values $Q^{(j)}(0)$ and $Q^{(j)}(1)$. In the case of degenerate boundary conditions and analytic $Q(\cdot)$, the criterion of completeness property is established.