Elastic scattering and bound states in the Aharonov-Bohm potential superimposed by an attractive ρ^-2potential

Abstract

We consider the elastic scattering and bound states of charged quantum particles moving in the Aharonov-Bohm and an attractive ρ −2 potential in a partial wave approach. Radial solutions of the stationary Schrödinger equation are specified in such a way that the Hamiltonian of the problem is self-adjoint. It is shown that they are not uniquely fixed but depend on open parameters. The related physical consequences are discussed. The scattering cross section is calculated and the energy spectrum of bound states is obtained.

A New Version of the Aharonov–Bohm Effect

“…For the case 3/4 < α < 1, one has α ∈ (b p , a p ] with p = 0, and for κ ∈ (− √ 1 − α, √ 1 − α)+ Z the operator H κ,α has deficiency indices n ± = 1, and n ± = 0 otherwise. This means that, if 0 ≤ κ < √ 1 − α, then the operator H κ,α has a one parameter family of self-adjoint extensions and a boundary condition is necessary to specify each of them; these boundary conditions are present in the domain of the operator (6). For √ 1 − α ≤ κ ≤ 1/2, the operator has only one self-adjoint extension and no boundary condition is needed; in this case, the self-adjoint extension is the operator closure of the initial Hamiltonian.…”

“…2. In [6], another set-up was proposed that is also modelled by the operator (2). It is the motion of a particle in the usual AB potential which is superimposed by a scalar potential U(r) = α r 2 (i.e., our V α ) of an electrically charged string in the direction of the solenoid.…”

Exact Bound State Solutions of the Schrödinger Equation for Noncentral Potential via the Nikiforov-Uvarov Method

“…In spite of this fact, some classes of noncentral potentials in three dimensions are solvable so long as the Schrödinger, Klein-Gordon and Dirac equations [15][16][17][18][19][20][21][22][23][24][25][26][27] with these potentials satisfy the separation of variables. These are also applicable to the scattering and condensed matter processes [28][29][30][31]. There have been many studies involving the potentials by using the well-known techniques, i.e., group theoretical manner [12,[32][33][34], supersymmetric formalism [35][36][37][38][39][40][41][42], path integral method [43][44][45][46][47][48][49][50] and other algebraic approaches [51][52][53][54][55][56][57][58][59][60].…”

Quantum motion in superposition of Aharonov–Bohm with some additional electromagnetic fields

“…It should be noted that exact solutions of the Schrödinger, Klein-Gordon and Dirac equations with the AB field in combination with the Coulomb field and the magnetic monopole field were studied in [23,24,25,26,27,13]. Exact solutions of the above mentioned equations with the AB field in combination with some other electromagnetic fields were presented in [28,13,29].The aim of the present work is to find the structure of the additional electromagnetic fields, for which the Schrödinger, Klein-Gordon, and Dirac equations can be solved exactly (in what follows, we call such fields exactly solvable additional fields), and to describe the corresponding exact solutions.…”