On two-dimensional Dirac operators with critical delta-shell interactions

We study two-dimensional Dirac operators with singular interactions of electrostatic and Lorentz-scalar type, supported either on a straight line or a circle. For certain critical values of the interaction strengths, the essential spectrum of such operators comprises an isolated point lying within the mass gap. We clarify the nature of this point in both geometries. For the straight line model, this point is known to be an eigenvalue of infinite multiplicity, and we provide a detailed analysis of the corresponding eigenfunctions. By contrast, in the case of a circle, we show that the said point is not itself an eigenvalue, but rather an accumulation point of a double sequence of simple eigenvalues. In view of the high degree of symmetry of the configurations under analysis, this behavior is unexpected and our findings lead us to formulate some conjectures concerning critical singular interactions supported on generic smooth curves.