2005/01/31 by D. R. Yafaev, Yafaev, D. R.
Mathematics · Physics and Astronomy · #47A40 #81U05 #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.FA #math.SP #msc:47A40 #msc:81U05
paper · pdf · doi:10.48550/arxiv.math/0501544
arxiv created 2005/01/31 · openalex publication_date 2005/01/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the scattering amplitude s(ω,ω^′;λ), ω,ω^′∈\Bbb Sd-1, λ> 0, corresponding to an arbitrary short-range magnetic field B(x), x∈\Bbb Rd. This is a smooth function of ω and ω^′ away from the diagonal ω=ω^′ 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 =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^′. 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).