The Riesz basis property with brackets for Dirac systems with summable potentials

A. M. Savchuk ; I. V. Sadovnichaya

Contemporary Mathematics. Fundamental Directions, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 1, Tome 58 (2015), pp. 128-152

Résumé

In the space $\mathbb H=(L_2[0,\pi])^2$, we study the Dirac operator $\mathcal L_{P,U},$ generated by the differential expression $\ell_P(\mathbf y)=B\mathbf y'+P\mathbf y$, where $$ B=\begin{pmatrix} -i0\\ 0 \end{pmatrix}, \qquad P(x)= \begin{pmatrix} p_1(x) p_2(x)\\ p_3(x) p_4(x) \end{pmatrix}, \qquad \mathbf y(x)= \begin{pmatrix} y_1(x)\\ y_2(x) \end{pmatrix}, $$ and the 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 elements of the matrix $P$ are assumed to be complex-valued functions summable over $[0,\pi]$. We show that the spectrum of the operator $\mathcal L_{P,U}$ is discrete and consists of eigenvalues $\{\lambda_n\}_{n\in\mathbb Z}$, such that $\lambda_n=\lambda_n^0+o(1)$ as $|n|\to\infty$, where $\{\lambda_n^0\}_{n\in\mathbb Z}$ is the spectrum of the operator $\mathcal L_{0,U}$ with zero potential and the same boundary conditions. If the boundary conditions are strongly regular, then the spectrum of the operator $\mathcal L_{P,U}$ is asymptotically simple. We show that the system of eigenfunctions and associate functions of the operator $\mathcal L_{P,U}$ forms a Riesz base in the space $\mathbb H$ provided that the eigenfunctions are normed. If the boundary conditions are regular, but not strongly regular, then all eigenvalues of the operator $\mathcal L_{0,U}$ are double, all eigenvalues of the operator $\mathcal L_{P,U}$ are asymptotically double, and the system formed by the corresponding two-dimensional root subspaces of the operator $\mathcal L_{P,U},$ is a Riesz base of subspaces (Riesz base with brackets) in the space $\mathbb H$.