  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.
