Riesz basicity with parentheses for Dirac system with summable potential

TL;DR

This paper investigates the spectral properties of a Dirac operator with summable potential and regular boundary conditions, establishing Riesz basis properties and asymptotic behaviors of eigenvalues.

Contribution

It proves the Riesz basis property for eigen and associated functions of Dirac operators with summable potentials under various boundary conditions.

Findings

Spectrum is purely discrete and asymptotically close to the zero-potential case.

Riesz basis property holds for strictly regular boundary conditions.

Full proof of Riesz basicity for non-strictly regular boundary conditions.

Abstract

We deal with the Dirac operator $\mathcal L_{P,U}$ generated in the space $\mathbb H=(L_2[0,\pi])^2$ by differential expression \begin{gather*} \ell_P(\mathbf y)=B\mathbf y'+P\mathbf y,\quad B = \begin{pmatrix} -i & 0 \\ 0 & i \end{pmatrix}, \qquad P(x) = \begin{pmatrix} p_1(x) & p_2(x) \\ p_3(x) & p_4(x) \end{pmatrix}, \qquad \mathbf y(x)=\begin{pmatrix}y_1(x)\\ y_2(x)\end{pmatrix}, \end{gather*} and regular boundary conditions $$ U(\mathbf y)=\begin{pmatrix}u_{11} & u_{12}\\ u_{21} & u_{22}\end{pmatrix}\begin{pmatrix}y_1(0)\\ y_2(0)\end{pmatrix}+\begin{pmatrix}u_{13} & u_{14}\\ u_{23} & u_{24}\end{pmatrix}\begin{pmatrix}y_1(\pi)\\ y_2(\pi)\end{pmatrix}=0. $$ The entries of a matrix $P$ suppose to be summable on the segment $[0,\pi]$ complex-valued functions. It is proved, that the operator $\mathcal L_{P,U}$ has purely discrete spectrum $\{\lambda_n\}_{n\in\mathbb Z}$ and $\lambda_n=\lambda_n^0+o(1)$ as $|n|\to\infty$. Here $\{\lambda_n^0\}_{n\in\mathbb Z}$ be the spectrum of operator $\mathcal L_{0,U}$ with zero potential and the same boundary conditions. In case this boundary conditions are strictly regular the spectrum of $\mathcal L_{P,U}$ is asymptotically simple. In this case the system of eigen and associated functions of operator $\mathcal L_{P,U}$ forms Riesz basis in $\mathbb H$. In case of regular but not strictly regular boundary conditions all eigenvalues of the operator $\mathcal L_{0,U}$ have multiplicity equal to $2$. In this case we give full proof of Riesz basicity of corresponding two-dimensional root subspaces of the operator $\mathcal L_{0,U}$.