Riesz basicity with parentheses for Dirac system with summable potential

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 $\mathcal L_{0,U}$.