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 $L.$