On the basis property of the system of eigenfunctions and associated functions of a one-dimensional Dirac operator

We study a one-dimensional Dirac system on a finite interval. The potential (a $2\times 2$ matrix) is assumed to be complex-valued and integrable. The boundary conditions are assumed to be regular in the sense of Birkhoff. It is known that such an operator has a discrete spectrum and the system $\{\mathbf{y}_n\}_1^\infty$ of its eigenfunctions and associated functions is a Riesz basis (possibly with brackets) in $L_2\oplus L_2$. Our results concern the basis property of this system in the spaces $L_\mu\oplus L_\mu$ for $\mu\ne2$, the Sobolev spaces ${W_2^\theta\oplus W_2^\theta}$ for $\theta\in[0,1]$, and the Besov spaces $B^\theta_{p,q}\oplus B^\theta_{p,q}$.