Spectral transition for Dirac operators with electrostatic $\delta$-shell potentials supported on the straight line

In this note the two dimensional Dirac operator $A_\eta$ with an electrostatic $\delta$-shell interaction of strength $\eta\in\mathbb R$ supported on a straight line is studied. We observe a spectral transition in the sense that for the critical interaction strengths $\eta=\pm 2$ the continuous spectrum of $A_\eta$ inside the spectral gap of the free Dirac operator $A_0$ collapses abruptly to a single point.