Riesz bases consisting of root functions of 1D Dirac operators

Abstract

For one-dimensional Dirac operators \[ L y = i ( 1 a m p ; 0 0 a m p ; − 1 ) d y d x + v y , v = ( 0 a m p ; P Q a m p ; 0 ) , y = ( y 1 y 2 ) , Ly= i \begin {pmatrix} 1 & 0 \\ 0 & -1 \end {pmatrix} \frac {dy}{dx} + v y, \quad v= \begin {pmatrix} 0 & P \\ Q & 0 \end {pmatrix}, \;\; y=\begin {pmatrix} y_1 \\ y_2 \end {pmatrix}, \] 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 , π ] , C 2 ) . L^2 ([0,\pi ], \mathbb {C}^2). In particular, if the potential matrix v v is skew-symmetric (i.e., Q ¯ = − P \overline {Q} =-P ), or more generally if Q ¯ = t P \overline {Q} =t P for some real t ≠ 0 , t \neq 0, then there exists a Riesz basis that consists of root functions of the operator L . L.