A family of analytically solvable potential models for the one- and two-channel problems is considered within the Jost matrix approach. The potentials are chosen to be constant in the interior region and to have different asymptotic behavior (tails) at large distances. The migration of the $S$-matrix poles on the Riemann surface of the energy, caused by variations of the potential strength, is studied. It is demonstrated that the long-range ($\sim 1/r^2$) tails and Coulomb potential ($\sim 1/r$) cause an unusual behavior of the $S$-matrix poles. It is found that in the two-channel problem with the long-range potentials the $S$-matrix poles may appear at complex energies on the physical Riemann sheet. The Coulomb tail not only changes the topology of the Riemann surface, but also breaks down the so-called mirror symmetry of the poles in both the single-channel and the two-channel problems.

中文翻译：

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一些可解析的势能模型的Jost矩阵

在Jost矩阵方法中考虑了针对一通道和两通道问题的一系列可解析的潜在模型。选择电位在内部区域中是恒定的，并在远距离处具有不同的渐近行为（尾部）。的迁移$\mathrm{小号}$研究了势能变化引起的能量在黎曼面上的-矩阵极。事实证明，远程（$〜\mathrm{1个}/{\mathrm{[R}}^{\mathrm{2个}}$）尾巴和库仑势（$〜\mathrm{1个}/\mathrm{[R}$）导致 $\mathrm{小号}$-矩阵极点。发现在具有远距离电势的两通道问题中，$\mathrm{小号}$-矩阵极点可能在物理黎曼片上以复能出现。库仑尾不仅改变了黎曼曲面的拓扑，而且打破了单通道和双通道问题中极点的所谓镜像对称性。