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Proceedings of the Japan Academy Series A Mathematical Sciences

Aharonov–Bohm effect in resonances for scattering by three solenoids

Hideo Tamura

Published Apr 1, 2015• 1 citation
Publisher

Abstract

We study how the Aharonov–Bohm effect is reflected in the location of quantum resonances for scattering by three solenoids at large separation. We also discuss what happens in the case of four solenoids.
DOI
10.3792/pjaa.91.45

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We study the Aharonov-Bohm effect (the AB effect) in quantum resonances for scattering by three solenoids in two dimensions under the situation that the centers of the solenoids are placed almost in line.The resonances are shown to be generated near the real axis by the trapping trajectories when the centers are largely separated from one another.We analyze how the AB effect is reflected in the location of these resonances.The results are described by use of the backward scattering amplitude by one solenoid and depend on the ratio of the distances between the solenoids as well as on the magnetic fluxes.The method makes a full use of the information from the scattering system by one solenoid, which is known to be exactly solvable in quantum mechanics.We also discuss the case of four solenoids.
We study the Aharonov–Bohm (AB) effect through resonances for magnetic scattering in two dimensions. The scattering system consists of three scatterers, one bounded obstacle, and two scalar potentials with compact supports at large separation, where the obstacle is placed between two supports and the support of the magnetic field is completely shielded by the obstacle. The field does not influence particles from a classical mechanical point of view, but quantum particles are influenced by the corresponding vector potential which does not necessarily vanish outside the obstacle. This quantum phenomenon is called the AB effect. The resonances are shown to be generated near the real axis by the trajectories oscillating between two supports of the scalar potentials as the distances between the three scatterers go to infinity. The location is described in terms of the backward amplitudes for scattering by each of the scalar potentials and by the obstacle, and it depends heavily on the magnetic flux of the field.
The role of the Aharonov-Bohm effect is analyzed for the bound states of a charged particle in the superposition of two fields: A uniform magnetic field plus the vector potential of a solenoid. Exact solutions are given for a solenoid of zero size and the corrections due to finite size are discussed. It is shown that a series of new "betatron resonances" appear in the spectrum, dependent on the flux quantum number $f=\frac{e\ensuremath{\Phi}}{\mathrm{hc}}$. The fundamental betatron resonance remains narrow and unshifted for solenoid radii large enough to broaden and shift the cyclotron resonance.
The authors have found the scattering cross sections on two solenoids with opposite magnetic fluxes and on the toroidal solenoid in the presence or absence of a magnetic field. From this they extract the diffraction pattern shift which is due to the presence of the magnetic flux inside the solenoids. Practical recommendations for the performance of the experiments are given. It is proved that the crucial experiment on the existence of the AB effect suggested earlier could indeed be realised in practice.
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.