Dirac fermions in a magnetic-solenoid field

TL;DR

This paper analyzes the Dirac equation in a magnetic-solenoid field, constructing self-adjoint Hamiltonians, exploring their spectra, and examining the effects of field regularization and boundary conditions.

Contribution

It introduces a comprehensive method for constructing self-adjoint Dirac Hamiltonians in magnetic-solenoid fields, including regularized cases, and provides explicit spectral analysis.

Findings

Constructed one- and two-parameter families of self-adjoint Hamiltonians.

Derived spectra and eigenfunctions for all extension parameters.

Presented an alternative method for self-adjoint extension construction.

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.