Hamiltonian self-adjoint extensions for (2+1)-dimensional Dirac particles

TL;DR

This paper analyzes the self-adjoint extensions of the Hamiltonian for a (2+1)-dimensional Dirac particle in a magnetic field with a flux tube, revealing how boundary conditions affect the spectrum and domain of the operator.

Contribution

It characterizes the self-adjoint extensions of the Dirac Hamiltonian in a magnetic flux, especially at critical angular momentum, and determines their spectral properties.

Findings

For l ≠ integer part of kappa, H_l is essentially self-adjoint.

For l = integer part of kappa, H_l admits a one-parameter family of self-adjoint extensions.

The spectrum depends on the extension parameter gamma and flux kappa.

Abstract

We study the stationary problem of a charged Dirac particle in (2+1) dimensions in the presence of a uniform magnetic field B and a singular magnetic tube of flux Phi = 2 pi kappa/e. The rotational invariance of this configuration implies that the subspaces of definite angular momentum l+1/2 are invariant under the action of the Hamiltonian H. We show that, for l different from the integer part of kappa, the restriction of H to these subspaces, H_l is essentially self-adjoint, while for l equal to the integer part of kappa, H_l admits a one-parameter family of self-adjoint extensions (SAE). In the later case, the functions in the domain of H_l are singular (but square-integrable) at the origin, their behavior being dictated by the value of the parameter gamma that identifies the SAE. We also determine the spectrum of the Hamiltonian as a function of kappa and gamma, as well as its closure.