Pauli Hamiltonians with an Aharonov-Bohm Flux
William Borrelli, Michele Correggi, Davide Fermi
TL;DR
We address the problem of a spin-\nobreak1/2 charged particle on the plane in an orthogonal AB flux, and provide a complete classification of all self-adjoint realizations of the 2D Pauli operator $H_{ m P}=(\bm{\sigma}\cdot(-i\nabla+\mathbf{A}))^2$. The main approach combines a Krein-type boundary-quadric framework with defect constructions to produce a resolvent formula $R_{ m P}^{(\Theta)}(z)=R_{ m P}^{(\mathrm{F})}(z)+\mathcal{G}(z)[\Lambda(z)+\Theta]^{-1}\widecheck{\mathcal{G}}(z)$, allowing explicit spectral and scattering characterizations: for all extensions the absolutely continuous spectrum is $\mathbb{R}^{+}$, while negative eigenvalues and zero-energy resonances occur according to $\det[\Lambda(-\mu)+\Theta]=0$ and $\det[\Lambda(0)+\Theta]=0$, respectively. The authors also establish completeness of wave operators, construct generalized eigenfunctions and scattering amplitudes, and analyze symmetries; they further relate Pauli extensions to Dirac extensions via $H_{ m P}=H_{ m D}^{2}$ and identify when a Pauli extension is the square of a Dirac extension, highlighting two scale-covariant Dirac cases. These results provide a rigorous, explicit framework for AB flux with spin, clarifying spectral, scattering, and symmetry structures and illuminating the Dirac-Pauli correspondence in singular magnetic fields.
Abstract
We study a two-dimensional Pauli operator describing a charged quantum particle with spin $1/2$ moving on a plane in presence of an orthogonal Aharonov-Bohm magnetic flux. We classify all the admissible self-adjont realizations and give a complete picture of their spectral and scattering properties. Symmetries of the resulting Hamiltonians are also discussed, as well as their connection with the Dirac operator perturbed by an Aharonov-Bohm singularity.