多通道散射本征相移和时延中Beutler-Fano公式的几何实现
Geometrical Realization of Beutler-Fano formulas appearing in eigenphase shifts and time delays in multichannel scattering
摘要：
最近，我们表明，本征相移和本征时间延迟附近的一个系统的一个离散状态和两个连续共振的Beutler-Fano公式的泛函使用适当的无量纲能量单位和线轮廓指数和确定的参数负责避免交叉的本征相移和本征时间延迟，也确定的参数负责本征时间延迟由于帧变换的变化。本文在Pauli矩阵所张成的三维Liouville空间中考虑了Beutler-Fano公式的几何实现，其中动力学算子是向量。对应于背景散射矩阵、S矩阵和时间延迟矩阵Q的矢量形成球面三角形，该球面三角形的顶角和顶角是与这些矩阵的本征通道之间的坐标系变换和散射矩阵的本征相移以及由于共振散射引起的相移有关的参数。球面三角形的余切定律产生Beutler-Fano共振公式，出现在本征相移和时间延迟。球面三角形的对偶性解释了参数之间关系的对称性，并提供了一个系统的方法来定义共轭动力学参数。球面三角形还显示了将背景散射中的通道-通道耦合与共振相互作用相结合的规则，从而在作为能量的函数的本征相移曲线中给出了避免交叉的相互作用。本文将前人和本文所发展的理论应用于三原子范德华分子的振动预离解。
英文摘要：
Recently, we showed that eigenphase shifts and eigentime delays near a resonance for a system of one discrete state and two continua are functionals of the Beutler-Fano formula using appropriate dimensionless energy units and line profile indices and identified parameters responsible for the avoided crossing of eigenphase shifts and eigentime delays and also identified parameters responsible for the eigentime delays due to a change in frame transformation. In this paper, the geometrical realization of the Beutler-Fano formulas is considered in the three-dimensional Liouville space spanned by the Pauli matrices, where dynamic operators are vectors. Vectors corresponding to the background scattering matrix, the S matrix, and the time delay matrix Q form a spherical triangle whose vertex and edge angles are parameters pertaining to the frame transformations among eigenchannels of those matrices and eigenphase shifts of the scattering matrices and the phase shift due to a resonance scattering. The cotangent laws of the spherical triangle yield Beutler-Fano resonance formulas appearing in eigenphase shifts and time delays. Duality holding for the spherical triangle explains the symmetry observed in the relations among parameters and provides a systematic way of defining conjugate dynamic parameters. The spherical triangle also shows the rule of combining the channel-channel couplings in the background scattering with the resonant interaction to give the avoided crossing interactions in the curves of eigenphase shifts as functions of eneryg. The theory developed in the previous and present papers is applied to the vibrational predissociation of triatomic van der Waals molecules.
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