Green functions of the Dirac equation with magnetic-solenoid field

Abstract

Various Green functions of the Dirac equation with a magnetic-solenoid field ͑the superposition of the Aharonov-Bohm field and a collinear uniform magnetic field͒ are constructed and studied. The problem is considered in 2ϩ1 and 3ϩ1 dimensions for the natural extension of the Dirac operator ͑the extension obtained from the solenoid regularization͒. Representations of the Green functions as proper time integrals are derived. The nonrelativistic limit is considered. For the sake of completeness the Green functions of the Klein-Gordon particles are constructed as well.

On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk

“…Since the self-adjoint extensions and the spectrum of the corresponding Dirac hamiltonian have already been discussed in [12] (see also [6] for the zero-field case), we will not dwell much on this point. One-vortex Green function was also computed in a closed form by Gavrilov et al [14]. However, the representation found in [14] is inconvenient for our purposes, so below we obtain another formula, in which the vortex-dependent contribution to the resolvent is manifestly separated from the "free" part.…”

“…One-vortex Green function was also computed in a closed form by Gavrilov et al [14]. However, the representation found in [14] is inconvenient for our purposes, so below we obtain another formula, in which the vortex-dependent contribution to the resolvent is manifestly separated from the "free" part. This formula enables us to find Fredholm determinant representations for the two-point tau function of the Dirac hamiltonian on the plane, which turns out to be related to a class of Painlevé V transcendents.…”

Zero Modes in a System of Aharonov–bohm Fluxes

“…In the relativistic case, the problem of defining the appropriate Dirac operator is discussed e.g. in [51,52,53], and at the presence of a uniform component -in the recent articles [54,55,56,57]. In all the mentioned papers, the spectral or scattering properties of the derived Hamiltonians are studied as well.…”

Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

“…We follow the notation of [5] for (2 + 1) γ-matrices: 8) and σ 1 , σ 2 , σ 3 are the Pauli spin matrices 2 . The four matrices…”

“…The topical problem in relativistic quantum theory is the self-adjoint extension of the Dirac Hamiltonian in external singular potentials. The Dirac equation for a magnetic solenoid field is the basis of the theory of the Aharonov-Bohm effect both in (3 + 1) and in (2 + 1) dimensions [6][7][8]. Khalilov [9] considered the Dirac equation in (2 + 1) dimensions for a relativistic charged zero-mass fermion in Coulomb and Aharonov-Bohm potentials in the context of a self-adjoint extension problem.…”