Zero-energy states of fermions in the field of Aharonov--Bohm type in 2+1 dimensions

TL;DR

This paper investigates the quantum behavior of fermions in a 2+1 dimensional Aharonov--Bohm field, deriving self-adjoint Hamiltonians, analyzing bound states, and exploring implications for fermion-antifermion pair creation.

Contribution

It provides a comprehensive analysis of self-adjoint extensions of the Dirac Hamiltonian with spin in an Aharonov--Bohm field, including boundary conditions and physical implications.

Findings

Existence of bound fermionic states under certain parameters.

Zero-energy fermion states indicate system instability.

Singular solutions arise from spin-magnetic field interactions.

Abstract

The quantum-mechanical problem of constructing a self-adjoint Hamiltonian for the Dirac equation in an Aharonov--Bohm field in 2+1 dimensions is solved with taking into account the fermion spin. The one-parameter family of self-adjoint extensions is found for the above Dirac Hamiltonian with particle spin. The correct domain of the self-adjoint Hamiltonian extension selecting by means of acceptable boundary conditions can contain regular and singular (at the point ${\bf r}=0$) square-integrable functions on the half-line with measure $rdr$. We argue that the physical reason of the existence of singular solutions is the additional attractive potential, which appear due to the interaction between the spin magnetic moment of fermion and Aharonov--Bohm magnetic field. For some range of parameters there are bound fermionic states. It is shown that fermion (particle and antiparticle) states with zero energy are intersected what signals on the instability of quantum system and the possibility of a fermion-antifermion pair creation by the static external field.