Eur. Phys. J. D 62, 361-370 (2011)
https://doi.org/10.1140/epjd/e2011-10726-y

Regular Article

On the solution of a “2D Coulomb + Aharonov-Bohm” problem: oscillator strengths in the discrete spectrum and scattering

Theoretical Physics Department, Voronezh State University, Universitetskaya sq.1, 394006 Voronezh, Russia

^a e-mail: melechp@yandex.ru

Received: 25 December 2010
Received in final form: 23 March 2011
Published online: 17 May 2011

Abstract

In this paper we present an exact analytic solution of the Schrödinger equation both in the discrete and continuous spectra for the combination of a 2D Coulomb potential and the Aharonov-Bohm flux. We analyze the influence of the Aharonov-Bohm flux on the energy spectrum of such a system and show that its presence leads to the broadening of the electron density in the bound states with the given value of the principal quantum number. We have shown that the scattering phase shift, which determines the S-matrix, can be represented as a sum of the Aharonov-Bohm scattering phase, first obtained by Henneberger, and a “modified” 2D Coulomb phase. We have noticed, that the Aharonov-Bohm scattering phase has a full analogy with the “quantum defect” for such a system. We have shown also, that the presence of the Aharonov-Bohm flux affects the radiation spectrum of the electron in this case, and this fact is demonstrated by calculations of the corresponding oscillator strengths. The explicit analytic expression for the scattering cross section on such a system is found in the frame of the eikonal approach. Obtained formula contains the two exact limiting cases, namely, the “pure” 2D Coulomb scattering as well as the “pure” Aharonov-Bohm effect. The mutual influence of a 2D Coulomb potential and the Aharonov-Bohm flux is also discussed.