环形线圈的散射
Scattering by a toroidal coil
摘要：
在本文中，我们考虑薛定谔算子在${\mathbb R}^3$与长程磁势相关的磁场内支持的圆环${\mathbb{T}}$。利用光滑扰动方法构造了稳态修正波算子和相应的散射矩阵$S(λ)$。我们证明了$S(λ)$的本质谱是单位圆的一个区间，它只取决于穿过$\mathbb{T}$截面的磁通量$ϕ$.此外，我们表明，在对比的Aharonov-Bohm势在${\mathbb{R}}^2$，总散射截面总是有限的。我们还推测，这里处理的情况下是一个典型的例子在3维。
英文摘要：
In this paper we consider the Schrödinger operator in ${\mathbb R}^3$ with a long-range magnetic potential associated to a magnetic field supported inside a torus ${\mathbb{T}}$. Using the scheme of smooth perturbations we construct stationary modified wave operators and the corresponding scattering matrix $S(λ)$. We prove that the essential spectrum of $S(λ)$ is an interval of the unit circle depending only on the magnetic flux $ϕ$ across the section of $\mathbb{T}$. Additionally we show that, in contrast to the Aharonov-Bohm potential in ${\mathbb{R}}^2$, the total scattering cross-section is always finite. We also conjecture that the case treated here is a typical example in dimension 3.
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