Consider the two dimensional Dirac operator with a real-valued constant Lorentz scalar $\delta$-shell interaction supported on a broken line. The self-adjointness of this operator was established recently by Frymark and Lotoreichik. In this talk the spectral properties of this model are discussed. The essential spectrum coincides with the essential spectrum of the Dirac operator with a $\delta$-interaction supported on the straight line, and it is known that the discrete spectrum of the latter operator is empty. However, it turns out that for a sufficiently small angle of the broken line the discrete spectrum is always non-empty. It is the first time that the existence of geometrically induced discrete eigenvalues is shown in the context of Dirac operators with singular potentials.
This talk is based on a joint work with D. Frymark and V. Lotoreichik.