On the discrete spectrum of Dirac operators with Lorentz-scalar $\delta$-shell interactions supported on unbounded curves

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $\delta$-shell interaction of strength $\tau \in (-\mathrm{\infty },0)\setminus \{-2\}$ supported on a $C^{\mathrm{\infty }}$-smooth curve $\mathrm{\Sigma }\subset {\mathbb{R}}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $\mathrm{\Sigma }$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $\mathrm{\Sigma }$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $\tau$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.