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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Scattering by magnetic fields

Author: D. R. Yafaev
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 875-895
MSC (2000): Primary 47A40, 81U05
Published electronically: July 27, 2006
MathSciNet review: 2241430
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Abstract: Consider the scattering amplitude $ s(\omega,\omega^\prime;\lambda)$, $ \omega,\omega^\prime\in{\mathbb{S}}^{d-1}$, $ \lambda > 0$, corresponding to an arbitrary short-range magnetic field $ B(x)$, $ x\in{\mathbb{R}}^d$. This is a smooth function of $ \omega$ and $ \omega^\prime$ away from the diagonal $ \omega=\omega^\prime$, but it may be singular on the diagonal. If $ d=2$, then the singular part of the scattering amplitude (for example, in the transversal gauge) is a linear combination of the Dirac $ \delta$-function and a singular denominator. Such a structure is typical of the long-range magnetic scattering. This phenomenon is referred to as the long-range Aharonov-Bohm effect. On the contrary, for $ d=3$ scattering is essentially of a short-range nature, although, for example, the magnetic potential $ A^{\mathrm{(tr)}}(x)$ such that $ \operatorname{curl} A^{\mathrm{(tr)}}(x)=B(x)$ and $ \langle A^{\mathrm{(tr)}}(x),x\rangle=0$, decays at infinity as $ \vert x\vert^{-1}$ only. More precisely, it is shown that, up to the diagonal Dirac function (times an explicit function of $ \omega$), the scattering amplitude has only a weak singularity in the forward direction $ \omega = \omega^\prime$. The approach is based on the construction (in the dimension $ d=3$) of a short-range magnetic potential $ A (x)$ corresponding to a given short-range magnetic field $ B(x)$.

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Additional Information

D. R. Yafaev
Affiliation: IRMAR, Université Rennes-I, Campus Beaulieu, 35042 Rennes Cedex, France

Keywords: Magnetic fields, scattering matrix, gauge transformations, long-range Aharonov--Bohm effect
Received by editor(s): January 20, 2005
Published electronically: July 27, 2006
Dedicated: Dedicated to the memory of Ol$^{′}$ga Aleksandrovna Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society