Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits
Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits
复制标题
狄拉克算子散射相位的渐近：高能、半经典和非相对论极限
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
D. Robert
中科院分区：
文献类型：
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作者：
V. Bruneau;D. Robert
在本文中，我们证明了欧几里德空间中狄拉克算子与电磁场扰动相关的散射相位（谱移函数）的几个结果。现在，在高能区域或半经典区域中，对于薛定谔算子的扰动，可以获得许多准确的结果。在这里，我们将这些结果扩展到狄拉克算子。由于狄拉克算子是一个系统，它的符号是一个4×4矩阵，并且它的连续谱有正值和负值，因此有几个技术问题需要克服。我们证明我们可以分离正能量和负能量来证明高能渐近展开，并在半经典情况下构造一个半经典 Foldy-Wouthuysen 变换。我们还证明了当光速趋于无穷大（非相对论极限）时散射相的渐近展开。
In this paper we prove several results for the scattering phase (spectral shift function) related with perturbations of the electromagnetic field for the Dirac operator in the Euclidean space.Many accurate results are now available for perturbations of the Schrödinger operator, in the high energy regime or in the semi-classical regime. Here we extend these results to the Dirac operator. There are several technical problems to overcome because the Dirac operator is a system, its symbol is a 4×4 matrix, and its continuous spectrum has positive and negative values. We show that we can separate positive and negative energies to prove high energy asymptotic expansion and we construct a semi-classical Foldy-Wouthuysen transformation in the semi-classical case. We also prove an asymptotic expansion for the scattering phase when the speed of light tends to infinity (non-relativistic limit).