Characterization of the potential smoothness of one-dimensional Dirac operator subject to general boundary conditions and its Riesz basis property

The one-dimensional Dirac operator with periodic potential $V=\begin{pmatrix} 0 & $\mathcal{P}$(x) \ $\mathcal{Q}$(x) & 0 \end{pmatrix}$,where$$\mathcal{P}$,$\mathcal{Q}$\in L^2([0,\pi])$ subject to periodic, antiperiodic or a general strictly regular boundary condition $(bc)$ has discrete spectrums. It is known that, for large enough $\mathrm{\mid }n\mathrm{\mid }$ in the disc centered at $n$ of radius 1/4, the operator has exactly two (periodic if $n$ is even or antiperiodic if $n$ is odd) eigenvalues ${\lambda }_n^{+}$ and ${\lambda }_n^{-}$ (counted according to multiplicity) and one eigenvalue ${\mu }_n^{bc}$ corresponding to the boundary condition $(bc)$. We prove that the smoothness of the potential could be characterized by the decay rate of the sequence $\mathrm{\mid }{\delta }_n^{bc}\mathrm{\mid }+\mathrm{\mid }{\gamma }_n\mathrm{\mid }$, where ${\delta }_n^{bc}={\mu }_n^{bc}-{\lambda }_n^{+}$ and ${\gamma }_n={\lambda }_n^{+}-{\lambda }_n^{-}\mathrm{.}$ Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if $\sup\limits_{\gamma_n\neq0} \frac{|\delta_n^{bc}|}{|\gamma_n|}$ is finite.