Non-relativistic limit of Dirac Hamiltonians with Aharonov-Bohm fields

TL;DR

This paper rigorously analyzes how Dirac operators with Aharonov-Bohm fields transition to Schrödinger operators in the non-relativistic limit, preserving key scattering properties and clarifying their mathematical relationship.

Contribution

It characterizes the non-relativistic limit of Dirac-AB operators, connecting them to Schrödinger-AB Hamiltonians while preserving scattering length and boundary conditions.

Findings

The non-relativistic limit maps Dirac-AB operators to Schrödinger-AB Hamiltonians.

The scattering length of the Aharonov-Bohm interaction is preserved in the limit.

The family of Dirac-AB operators corresponds to a physically relevant sub-family of Schrödinger-AB Hamiltonians.

Abstract

We characterise the families of self-adjoint Dirac and Schr\"{o}dinger 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, Schr\"{o}\-dinger-AB Hamiltonians with relativistic Dirac approximants.