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

TL;DR

This paper characterizes the smoothness of one-dimensional Dirac operators with various boundary conditions through spectral decay rates and establishes conditions for their Riesz basis property.

Contribution

It provides a new spectral characterization of potential smoothness and criteria for Riesz basis property in Dirac operators with general boundary conditions.

Findings

Spectral decay rates characterize potential smoothness.

Riesz basis property linked to boundedness of spectral ratios.

Results apply to operators with periodic, antiperiodic, and general boundary conditions.

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.