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

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 Φ = 2πκ/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 κ-l⩾1 or κ-l⩽0 the restriction of H to these subspaces, Hl, is essentially self-adjoint, while for 0<κ-l<1 Hl admits a one-parameter family of self-adjoint extensions (SAEs). In the latter case, the functions in the domain of Hl are singular (but square integrable) at the origin, their behaviour being dictated by the value of the parameter γ that identifies the SAE. We also determine the spectrum of the Hamiltonian as a function of κ and γ, as well as its closure.