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International Journal of Modern Physics A

Aharonov–Bohm effect from an attractive r−2 scalar potential together with a r−1 scalar potential

J. Carvalho Veloso, K. Bakke

Published Oct 23, 2025• 0 citations
Publisher

Abstract

By confining a point charge to an attractive [Formula: see text] potential together with a [Formula: see text] scalar potential, we study the Aharonov–Bohm effect for bound states. In view of the presence of the centrifugal term and the attractive [Formula: see text] potential, we analyze the case where the attractive [Formula: see text] potential is the dominant term, and that one where the centrifugal term is the dominant term. In both cases, we show that the bound states arise around the inner radius of the long nonconducting cylinder.

DOI
10.1142/s0217751x25501842

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By confining a point charge to an attractive [Formula: see text] potential together with a [Formula: see text] scalar potential, we study the Aharonov–Bohm effect for bound states. In view of the presence of the centrifugal term and the attractive [Formula: see text] potential, we analyze the case where the attractive [Formula: see text] potential is the dominant term, and that one where the centrifugal term is the dominant term. In both cases, we show that the bound states arise around the inner radius of the long nonconducting cylinder.
We explore a mesoscopic system made of a long nonconducting cylinder of inner radius [Formula: see text] where a spinless electron interacts with an attractive [Formula: see text] potential. We analyze the asymptotic behavior of the radial wave function by assuming that there is an impenetrable potential wall located at [Formula: see text], and thus, discuss the effects of rotation on the energy levels, as well as the Aharonov–Bohm effect for bound states. Further, we include a non-Hermitian operator and show that the non-Hermitian Hamiltonian has real energy eigenvalues. We also discuss the effects of rotation and the Aharonov–Bohm effect from the real energy eigenvalues. We extend our discussion to the revival time.
An asymptotic formula for the energy eigenvalues of a charged particle contained in a cylindrical shell is derived. The formula is used to estimate the eigenvalues when there is a cylindrically symmetric static magnetic-induction field present and the system exhibits the bound-state Aharonov-Bohm effect. It is shown that the errors involved form an sequence with terms labelled by the radial quantum number. Numerical approximations obtained from the asymptotic formula are compared to those obtained from analytical solutions of Schrödinger's equation in the following two cases. (i) There is no magnetic-induction field in the shell itself but there is a field inside the inner cylinder. (ii) There is a constant magnetic-induction field in the shell as well as a (possibly different) field inside the inner cylinder. In these cases, the radial eigenfunctions are expressible in terms of Bessel functions and confluent hypergeometric functions, respectively. The numerical results are consistent with the asymptotic character of the formula for the energy eigenvalues.
The authors consider a charged particle confined to a ring encircling a magnetic flux. Supposing that the particle is trapped by a delta-function potential located on the ring, they examine the flux dependence of the energy and the wavefunction of the bound state. This offers a simple explicit model for the Aharonov-Bohm effect on a system localized, but not completely confined, to a simply connected region off a magnetic flux. The result is used to show that Berry's phase for the bound state, when the delta-function potential is slowly transported around the ring, consists not only of the Aharonov-Bohm phase proportional to the flux, but also of an extra term which reflects the incomplete localization of the state; in the limit of strong binding the extra term depends sinusoidally on the flux.
The ((2+1)-dimensional) Aharonov–Bohm effect is analyzed for a spin-1/2 particle in the case that a 1/rpotential is present. Scalar and vector couplings are each considered. It is found that the approach in which the flux tube is given a finite radius that is taken to zero only after a matching of boundary conditions does not give physically meaningful results. Specifically, the operations of taking the limit of zero flux tube radius and the Galilean limit do not commute. Thus there appears to be no satisfactory solution of the relativistic Aharonov–Bohm–Coulomb problem using the finite radius flux tube method.