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Journal of the London Mathematical Society 2006 74(3):695-716; doi:10.1112/S0024610706023325
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© The London Mathematical Society

Semiclassical Analysis for Magnetic Scattering by Two Solenoidal Fields

Hiroshi T. Ito and Hideo Tamura

Department of Computer Science, Ehime University Matsuyama 790-8577, Japan ito{at}cs.ehime-u.ac.jp
Department of Mathematics, Okayama University Okayama 700-8530, Japan tamura{at}math.okayama-u.ac.jp

Received 30 June 2005. Revision received 21 February 2006.

That vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the A–B (Aharonov–Bohm) effect. We study scattering by two solenoidal magnetic fields (point-like magnetic fields) in two dimensions and analyze the asymptotic behavior of the scattering amplitude in the semiclassical limit. The corresponding classical mechanical system has a trajectory oscillating between the centers of the two fields. We derive the asymptotic formula with the first three terms and make clear how such a trapping trajectory is reflected in the asymptotic formula through the A–B effect. We also make a brief comment on an extension to the scattering by many solenoidal fields. The result depends on the location of the centers of the fields. In particular, the A–B effect appears strongly when the centers are on an even line.


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