Riesz bases consisting of root functions of 1D Dirac operators

Abstract: For one-dimensional Dirac operators $$ Ly= i \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} \frac{dy}{dx} + v y, \quad v= \begin{pmatrix} 0 & P \ Q & 0 \end{pmatrix}, \;\; y=\begin{pmatrix} y_1 \ y_2 \end{pmatrix}, $$ subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in $L^2 ([0,\pi], \mathbb{C}^2).$ In particular, if the potential matrix $v$ is skew-symmetric (i.e., $\overline{Q} =-P$), or more generally if $\overline{Q} =t P$ for some real $t \neq 0,$ then there exists a Riesz basis that consists of root functions of the operator $L.$