We describe the self-adjoint realizations of the operator H :We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.

Under certain hypothesis of smallness of the regular potential V, we prove that the Dirac operator in R 3 coupled with a suitable rescaling of V converges in the strong resolvent sense to the Hamiltonian coupled with a δ-shell potential supported on Σ, a bounded C 2 surface. Nevertheless, the coupling constant depends non-linearly on the potential V: the Klein's Paradox comes into play.

This note revolves on the free Dirac operator in R 3 and its δ-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich, Exner and Šeba [Dirac operators with a spherically symmetric δ-shell interaction, J. Math. Phys. 30.12 (1989), 2875-2882] and by Arrizabalaga, Mas and Vega [Shell interactions for Dirac operators, J. Math. Pures et Appl. 102.4 (2014), 617-639] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case.

We investigate the self-adjointness of the two-dimensional Dirac operator D, with quantum-dot and Lorentz-scalar ı-shell boundary conditions, on piecewise C 2 domains (with finitely many corners). For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space H 1=2 , the formal form domain of the free Dirac operator. The main part of our paper consists of a description of the domain of the adjoint operator D in terms of the domain of D and the set of harmonic functions that verify some mixed boundary conditions. Then, we give a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.