关于S-矩阵的优美解析结构
On beautiful analytic structure of the S-matrix
摘要：
对于指数衰减的势函数，$s$波S矩阵的解析结构可以确定到最微小的细节，包括其所有极点及其留数的位置。美丽的隐藏结构可以通过其域着色来揭示。S-矩阵的一个基本性质是，任何束缚态都对应于复能量平面物理层上S-矩阵的一个极点。对于排斥指数衰减势，物理层上的$s$波S矩阵的无穷多个极点都不对应于任何物理状态。在复能量平面的第二片上，S-矩阵具有对应于虚态的无穷多个极点和对应于共振和反共振的互补对的有限多个极点。冗余极点和零点的来源被证实与两个线性无关的解析函数的一个参数的解析延拓的特性有关。冗余极点对渐近完备性关系的总体贡献，只要留数定理可以应用，被确定为振荡函数。
英文摘要：
For an exponentially decaying potential, analytic structure of the $s$-wave S-matrix can be determined up to the slightest detail, including position of all its poles and their residues. Beautiful hidden structures can be revealed by its domain coloring. A fundamental property of the S-matrix is that any bound state corresponds to a pole of the S-matrix on the physical sheet of the complex energy plane. For a repulsive exponentially decaying potential, none of infinite number of poles of the $s$-wave S-matrix on the physical sheet corresponds to any physical state. On the second sheet of the complex energy plane, the S-matrix has infinite number of poles corresponding to virtual states and a finite number of poles corresponding to complementary pairs of resonances and anti-resonances. The origin of redundant poles and zeros is confirmed to be related to peculiarities of analytic continuation of a parameter of two linearly independent analytic functions. The overall contribution of redundant poles to the asymptotic completeness relation, provided that the residue theorem can be applied, is determined to be an oscillating function.
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