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

Abstract

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 $|n|$ 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 $|\delta_n^{bc}|+|\gamma_n|$, where $\delta_n^{bc}=\mu_n^{bc}-\lambda_n^+$ and $\gamma_n=\lambda_n^+-\lambda_n^-.$ 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.