Scattering at Low Energies on Manifolds with Cylindrical Ends and Stable Systoles
Scattering at Low Energies on Manifolds with Cylindrical Ends and Stable Systoles
复制标题
具有圆柱形末端和稳定收缩的流形上的低能量散射
DOI：
10.1007/s00039-010-0079-2
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发表时间：
2009
影响因子：
2.2
通讯作者：
A. Strohmaier
中科院分区：
文献类型：
--
作者：
Werner Mueller;A. Strohmaier
具有圆柱端的流形上的p-形式的散射理论有一个直接的上同调解释。利用Hodge同构，低能散射矩阵可以看作是边界上的上同调算子。它的值为零描述了绝对上同调在边界的上同调中的图像。我们表明，所谓的散射长度，在零能量的艾森巴德-维格纳时间延迟，有一个上同调的解释。即在上同调的长正合列中，它将上同调类在边界上的范数与它的象在连通同态下的范数联系起来。一个有趣的结果是，这是一个可以估计散射长度的几何数据，如某些同调收缩的体积。
Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism, the scattering matrix at low energy may be regarded as an operator on the cohomology of the boundary. Its value at zero describes the image of the absolute cohomology in the cohomology of the boundary. We show that the so-called scattering length, the Eisenbud–Wigner time delay at zero energy, has a cohomological interpretation as well. Namely, it relates the norm of a cohomology class on the boundary to the norm of its image under the connecting homomorphism in the long exact sequence in cohomology. An interesting consequence of this is that one can estimate the scattering lengths in terms of geometric data like the volumes of certain homological systoles.