Dirac fermions in a magnetic-solenoid field

TL;DR

This work provides a rigorous, fully self-adjoint formulation of the Dirac equation for a charged particle in a magnetic-solenoid field, the superposition of an Aharonov–Bohm flux and a uniform magnetic field. By applying von Neumann's extension theory, the authors construct a one-parameter family of self-adjoint extensions in 2+1 dimensions and a two-parameter family in 3+1 dimensions, with domains fixed by asymptotic boundary conditions at the solenoid, and derive exact bound-state spectra and spinor eigenfunctions for all extension parameters. They also analyze a regularized finite-radius solenoid, deducing how the zero-radius limit fixes concrete boundary conditions (Θ) and clarifying the role of solenoid-core physics; a spinless case and a reduced, practical extension method for radial operators are also presented. The results elucidate how singular magnetic fluxes alter the self-adjointness, spectra, and boundary physics of Dirac fermions, with implications for AB-type phenomena in relativistic settings and for rigorous treatment of singular gauge fields.

Abstract

We consider the Dirac equation with a magnetic-solenoid field (the superposition of the Aharonov--Bohm solenoid field and a collinear uniform magnetic field). Using von Neumann's theory of the self-adjoint extensions of symmetric operators, we construct a one-parameter family and a two-parameter family of self-adjoint Dirac Hamiltonians in the respective 2+1 and 3+1 dimensions. Each Hamiltonian is specified by certain asymptotic boundary conditions at the solenoid. We find the spectrum and eigenfunctions for all values of the extension parameters. We also consider the case of a regularized magnetic-solenoid field (with a finite-radius solenoid field component) and study the dependence of the eigenfunctions on the behavior of the magnetic field inside the solenoid. The zero-radius limit yields a concrete self-adjoint Hamiltonian for the case of the magnetic-solenoid field. In addition, we consider the spinless particle in the regularized magnetic-solenoid field. By the example of the radial Dirac Hamiltonian with the magnetic-solenoid field, we present an alternative, more simple and efficient, method for constructing self-adjoint extensions applicable to a wide class of singular differential operators.

Figures (2)

• Figure 1: Particle lower energy levels in dependence on the parameter $\eta _{+}=\frac{\Gamma \left( \mu \right) }{\Gamma \left( 1-\mu \right) }\left( \frac{\gamma }{2M^{2}}\right) ^{1-\mu }\tan \left( \frac{\pi }{4}+\frac{\Theta }{2}\right)$
• Figure 2: Antiparticle lower energy levels in dependence on the parameter $\eta _{-}=\frac{\Gamma \left( \mu \right) }{\Gamma \left( 1-\mu \right) }\left( \frac{\gamma }{2M^{2}}\right) ^{-\mu }\tan \left( \frac{\pi }{4}+\frac{\Theta }{2}\right)$