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Annales Henri Poincaré

Aharonov–Bohm Effect in Resonances of Magnetic Schrödinger Operators with Potentials with Supports at Large Separation

Ivana Alexandrova, Hideo Tamura

Published Feb 24, 2011• 3 citations
Publisher
DOI
10.1007/s00023-011-0088-1

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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.
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.