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

Abstract

One dimensional Dirac operators $$ L_{bc}(v) \, y = i \begin{pmatrix} 1 & 0 0 & -1 \end{pmatrix} \frac{dy}{dx} + v(x) y, \quad y = \begin{pmatrix} y_1 y_2 \end{pmatrix}, \quad x\in[0,π],$$ 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 $λ_{k,α}^0 = k + τ_α$ where $ \; α\in \{1,2\}, \; k \in 2 \mathbb{Z} $ and $τ_α=τ_α(bc);$ if $|k|>N(v, bc) $ each of the discs $D_k^α= \{z: \; |z-λ_{k,α}^0| <ρ=ρ(bc) \} , $ $α\in \{1,2\}, $ contains exactly one simple eigenvalue $λ_{k,α} $ of $L_{bc} (v) $ and $(λ_{k,α} -λ_{k,α}^0)_{k\in 2\mathbb{Z}} $ is an $\ell^2 $-sequence. Moreover, it is proven that the root projections $ P_{n,α} = \frac{1}{2πi} \int_{\partial D^α_n} (z-L_{bc} (v))^{-1} dz $ satisfy the Bari--Markus condition $$\sum_{|n| > N} \|P_{n,α} - P_{n,α}^0\|^2 < \infty, \quad n \in 2\mathbb{Z}, $$ 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.