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

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

2010-08-24

arxiv.org/abs/1008.4095v1

Mathematics

Spectral Theory

Scientific paper

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.

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