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Advances in Mathematical Physics

Description of the Magnetic Field and Divergence of Multisolenoid Aharonov-Bohm Potential

Araz Rafig ogly Aliev, E. H. Eyvazov, S. F. M. Ibrahim, Hassan A. Zedan

Published Jan 1, 2016• 1 citation
Publisher

Abstract

Explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential are obtained; the mathematical essence of this potential is explained. It is shown that the magnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids. Deficiency index is found for the minimal operator generated by the Aharonov-Bohm differential expression.

DOI
10.1155/2016/8645732

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Explicit formulas for the magnetic field and divergence of multisolenoid Aharonov-Bohm potential are obtained; the mathematical essence of this potential is explained. It is shown that the magnetic field and divergence of this potential are very singular generalized functions concentrated at a finite number of thin solenoids. Deficiency index is found for the minimal operator generated by the Aharonov-Bohm differential expression.
We briefly review the scattering theory for the Schrödinger operator related to the Aharonov-Bohm effect, by exhibiting some explicitly solvable models; an infinitesimally thin solenoid in mathbb{R}^{2}, two infinitesimally thin solenoids with the quantized magnetic fluxes in mathbb{R}^{2}, and an infinitesimally thin ring solenoid with the quantized magnetic flux in mathbb{R}^{3}. We give explicit formulas for the incoming wave solution and the scattering amplitude for these models in terms of some special functions, with the graphs obtained by a numerical method. We also review known high-energy asymptotic formulas related to these examples.
We add a scalar potential to the 2D Aharonov-Bohm (AB) model which properly diverges both at the solenoid border and at infinity so that the resulting operator is essentially self-adjoint and has a discrete spectrum; the former property is interpreted as no contact of the particle with the solenoid border since there is no need of boundary conditions. We study gauge transformations to get the usual periodic behavior of the AB properties as a function of the magnetic flux. The presence of the AB effect is proven through the ground state energy, which is shown to be smooth in case it is simple and with a nonzero derivative if the ground state is real valued; such properties are verified in the case of circular solenoids, for which it is shown to be a nonconstant periodic function with a minimum at integer and a maximum at half-integer circulations (at half-integer circulations, it is doubly degenerated).
We consider a spin-1/2 charged particle in the plane under the influence of two idealized Aharonov–Bohm fluxes. We show that the Pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices. Further we explicitly construct a basis in the deficiency subspaces of the symmetric operator obtained by restricting the domain to functions with supports separated from the vortices. This construction makes it possible to apply the Krein’s formula to the Pauli operator.