article

Title: Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

Authors: Plamen Djakov and Boris Mityagin

Issue: Volume 61 (2012), Issue 1, 359-398

Abstract:

$One-dimensional Dirac operators \[ L_{bc}(v)y = i\begin{pmatrix}1&\hfill 0\0&-1\end{pmatrix}\frac{\mathrm{d}y}{\mathrm{d}x}+v(x)y,\quad y=\begin{pmatrix}y_1\y_2\end{pmatrix}, x\in[0,\pi], \] considered with $L^2$-potentials $v(x) = \left(\begin{smallmatrix}0&P(x)\\Q(x)&0\end{smallmatrix}\right)$ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc$, it is shown that every eigenvalue of the free operator $L^0_{bc}$ is simple and has the form $\lambda_{k,\alpha}^0 = k+\tau_{\alpha}$, where $\alpha\in\{1,2\}$, $k\in 2\mathbb{Z}$ and $\tau_{\alpha} = \tau_{\alpha}(bc)$; if $|k| > N(v,bc)$, each of the discs $D_k^{\alpha} = \{z : |z-\lambda_{k,\alpha}^0| < ho = \rho(bc)\}$, $\alpha\in\{1,2\}$, contains exactly one simple eigenvalue $\lambda_{k,\alpha}$ of $L_{bc}(v)$ and $(\lambda_{k,\alpha}-\lambda_{k,\alpha}^0)_{k\in 2\mathbb{Z}}$ is an $\ell^2$-sequence. Moreover, it is proven that the root projections $P_{n,\alpha} = 1/(2\pi i)\int_{\partial D^{\alpha}_n}(z-L_{bc}(v))^{-1}\mathrm{d}z$ satisfy the Bari-Markus condition \[ \sum_{|n| > N}\|P_{n,\alpha}-P_{n,\alpha}^0\|^2 < \infty,\quad n\in 2\mathbb{Z}, \] where $P_n^0$ are the root projections of the free operator $L^0_{bc}$. Hence, for strictly regular $bc$, there is a Riesz basis consisting of root functions (all but finitely many being eigenfunctions). Similar results are obtained for regular but not strictly regular $bc$---then in general there is no Riesz basis consisting of root functions, but we prove that the corresponding system of two-dimensional root projections is a Riesz basis of projections.$