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

TL;DR

This paper proves unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions, showing the existence of Riesz bases of root functions or projections depending on the regularity of the boundary conditions.

Contribution

It establishes spectral properties and basis convergence results for 1D Dirac operators with regular boundary conditions, including the case of strictly regular and not strictly regular boundary conditions.

Findings

Eigenvalues are simple and asymptotically close to free operator eigenvalues.

Root projections satisfy Bari–Markus condition, ensuring basis properties.

Existence of Riesz basis of root functions or projections depending on boundary condition regularity.

Abstract

One dimensional Dirac operators $$ L_{bc}(v) \, y = i \begin{pmatrix} 1 & 0 0 & -1 \end{pmatrix} \frac{dy}{dx} + v(x) y, \quad y = \begin{pmatrix} y_1 y_2 \end{pmatrix}, \quad x\in[0,\pi],$$ considered with $L^2$-potentials $ v(x) = \begin{pmatrix} 0 & P(x) Q(x) & 0 \end{pmatrix} $ 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| <\rho =\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} = \frac{1}{2\pi i} \int_{\partial D^\alpha_n} (z-L_{bc} (v))^{-1} dz $ 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.