Aharonov-Bohm effect without contact with the solenoid

Abstract

We add a confining potential to the Aharonov-Bohm model resulting in no contact of the particle with the solenoid (border); this is characterized by a unique self-adjoint extension of the initial Hamiltonian operator. It is shown that the spectrum of such extension is discrete and the first eigenvalue is found to be a nonconstant 1-periodic function of the magnetic flux circulation with a minimum at integers and maximum at half-integer circulations. This is a rigorous verification of the effect.

A New Version of the Aharonov–Bohm Effect

“…Yakir Aharonov and David Bohm [4] have argued that the (magnetic) potential can influence the physics of a quantum particle, even when the particle does not directly interact with the magnetic field (see [23] for an earlier, although more restricted, discussion). Such kind of influence has been verified in scattering operators and cross sections (see [4,34,17,2,33,8,19] to mention just some possible references) and, in some cases, in the Hamiltonian eigenvalues [31,26,21]. Of course such influence can be detected in other quantities derived from eigenvalues and/or scattering operators.…”

“…By applying a natural shielding method to the initial AB Hamiltonian [27,29,18,20], the Dirichlet boundary condition (i.e., wavefunctions vanish at the solenoid) is selected. In [21] we have recently proposed a modification of the AB Hamiltonian that is essentially self-adjoint, that is, a model for which there is exactly one self-adjoint extension; the physical interpretation is the absence of contact of the particle with the solenoid in this case. This was obtained by the introduction of an additional potential that conveniently diverges close to the solenoid.…”

“…The purpose of this note is to present a situation where, effectively, instead of scattering quantities or the eigenvalues of the Hamiltonian, the Aharonov-Bohm effect is probed by the number of self-adjoint extensions of the original energy operator. The basic point is to add a potential, similar to our approach in [21], but for solenoids of zero radius (such solenoids have been considered in the original paper by Aharonov and Bohm and in many other works; for instance in [2,17]). Although technically very simple, we have found a situation in which the AB magnetic potential is responsible for the values of the deficiency indices of the original Hamiltonian, a very strong influence that (to the best of our knowledge) has not been reported yet.…”

“…Another point is that the considered model makes use of the idealized zero radius solenoid, and this was (technically) important for us (compare with [21] that has considered nonzero radius solenoids). Although there are some worries about idealizations, in many instances they make some models mathematically tractable, whereas still keeping important aspects of real systems and help us to understand the foundations of the phys-ical theory; we reinforce that the original scattering calculations by Aharonov and Bohm have also used solenoids of zero radius.…”

Dynamic of Scalar Bosons in Aharonov-Bohm Magnetic Field

“…Notably, researchers [22] [23] [24] [25], in dealing with the initial Aharonov-Bohm Hamiltonian, employed the natural shielding method and opted for the Dirichlet boundary condition, wherein wave functions vanish at the solenoid. In a recent development, [26] proposed a modification of the AB Hamiltonian that is essentially self-adjoint, signifying…”