We consider the Dirac operators with singular potentials 1 \(\begin{aligned} \mathfrak {D}_{m}=\sigma _{1}D_{x_{1}}+\sigma _{2}D_{_{x_{2}}}+\sigma _{3}m\,\ +Q\delta _{\Gamma } \end{aligned}\) where \(\sigma _{j},j=1,2,3\) are the Pauli matrices, m is a mass of the particle, \(Q\delta _{\Gamma }\) is a singular potential supported on a composite Alfors–David–Carleson curve with a finite set of nodes. We reduce interaction problems to corresponding singular integral equations on composite curves \(\Gamma \) . We prove that unbounded Dirac interaction operators are self-adjoint in the space \(L^{2}(\mathbb {R}^{2}, \mathbb {C}^{2})\) if the associated integral operators on \(\Gamma \) are Fredholm. We investigate the Fredholm properties of these operators and apply them to the study of spectral properties of interaction problems. We consider, as example, the interaction problems given by the singular potentials on \(\Gamma \) which are the sum of electrostatic and scalar Lorentz potentials.
