Riesz bases consisting of root functions of 1D Dirac operators

TL;DR

This paper establishes conditions under which the root functions of 1D Dirac operators form Riesz bases in L^2, especially for skew-symmetric or proportional potential matrices, enhancing spectral analysis tools.

Contribution

It provides necessary and sufficient conditions for Riesz bases of root functions in 1D Dirac operators, including specific cases with skew-symmetric or proportional potentials.

Findings

Riesz bases exist under certain potential conditions

Necessary and sufficient criteria are identified

Special cases include skew-symmetric and proportional potentials

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 $L.$