Scattering by magnetic fields

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dc.creatorYafaev, D. R.
dc.date2005-01-31
dc.date.accessioned2026-07-07T05:16:32Z
dc.date.available2026-07-07T05:16:32Z
dc.descriptionConsider the scattering amplitude $s(ω,ω^\prime;λ)$, $ω,ω^\prime\in{\Bbb S}^{d-1}$, $λ> 0$, corresponding to an arbitrary short-range magnetic field $B(x)$, $x\in{\Bbb R}^d$. This is a smooth function of $ω$ and $ω^\prime$ away from the diagonal $ω=ω^\prime$ 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 function and of a singular denominator. Such structure is typical for long-range scattering. We refer to this phenomenon as to the long-range Aharonov-Bohm effect. On the contrary, for $d=3$ scattering is essentially of short-range nature although, for example, the magnetic potential $A^{(tr)}(x)$ such that ${\rm curl} A^{(tr)}(x)=B(x)$ and $<A^{(tr)}(x),x>=0$ decays at infinity as $|x|^{-1}$ only. To be more precise, we show that, up to the diagonal Dirac function (times an explicit function of $ω$), the scattering amplitude has only a weak singularity in the forward direction $ω= ω^\prime$. Our approach relies on a construction in the dimension $d=3$ of a short-range magnetic potential $A (x)$ corresponding to a given short-range magnetic field $B(x)$.
dc.identifierhttps://arxiv.org/abs/math/0501544
dc.identifierhttp://arxiv.org/abs/math/0501544
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/74023
dc.subjectSpectral Theory
dc.subjectFunctional Analysis
dc.subject47A40;81U05
dc.titleScattering by magnetic fields
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