Non-Relativistic Limit of Dirac Hamiltonians With Aharonov–Bohm Fields

Abstract

We characterize the families of self-adjoint Dirac and Schrödinger operators with Aharonov–Bohm magnetic field, and we exploit the non-relativistic limit of infinite light speed to connect the former to the latter. The limit consists of the customary removal of the rest energy and of a suitable scaling, with the light speed, of the short-scale boundary condition of self-adjointness. This ensures that the scattering length of the Aharonov–Bohm interaction is preserved along the limit. Noteworthy is the fact that the whole family of Dirac-AB operators is mapped, in the non-relativistic limit, into the physically relevant sub-family of $s$-wave, angular-momentum-commuting, Schrödinger–AB Hamiltonians with relativistic Dirac approximants.