On the completeness property of root vector systems for 2 × 2 Dirac type operators with non-regular boundary conditions

Abstract

The paper is concerned with the completeness property of a system of root vectors of a boundary value problem for the following $2\times 2$ Dirac type equation$Ly=-i{B}^{-1}{y}^{\prime }+Q\left(x\right)y=\lambda y,y=\mathrm{col}(y_1,y_2),x\in [0,1],B=\mathrm{diag}(b_1,b_2),b_1<0<b_2,\text{and}Q\in W_1^n[0,1]\otimes C^{2\times 2},$ subject to general non-regular two-point boundary conditions $Cy\left(0\right)+Dy\left(1\right)=0$. If $b_2=-b_1=1$, this equation is equivalent to the one dimensional Dirac equation.

We establish a 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 values $Q^{\left(j\right)}\left(0\right)$ and $Q^{\left(j\right)}\left(1\right)$. In the case of degenerate boundary conditions and the analytic $Q(\cdot )$, the criterion of the completeness property is established. We demonstrate our results on the explicit example of a complete system of vector quasi-exponential polynomials.

Applications to the spectral synthesis for Dirac type operators are discussed. Moreover, applications to the completeness property for the damped string equation are provided.