In this followup article to Ducru et al. [Phys. Rev. C 103, 064608 (2021)], we establish new results on scattering matrix pole expansions for complex wave numbers in $R$-matrix theory. In the past, two branches of theoretical formalisms emerged to describe the scattering matrix in nuclear physics: $R$-matrix theory and pole expansions. The two have been quite isolated from one another. Recently, our study of Brune's alternative parametrization of $R$-matrix theory has shown the need to extend the scattering matrix (and the underlying $R$-matrix operators) to complex wave numbers. Two competing ways of doing so have emerged from a historical ambiguity in the definitions of the shift $\mathit{S}$ and penetration $\mathit{P}$ functions: the legacy Lane and Thomas's “force closure” approach versus analytic continuation (which is the standard in mathematical physics). The $R$-matrix community has not yet come to a consensus as to which to adopt for evaluations in standard nuclear data libraries, such as ENDF. In this article, we argue in favor of analytic continuation of $R$-matrix operators. We bridge $R$-matrix theory with the Humblet-Rosenfeld pole expansions, and discover new properties of the Siegert-Humblet radioactive poles and widths, including their invariance properties to changes in channel radii $a_c$. We then show that analytic continuation of $R$-matrix operators preserves important physical and mathematical properties of the scattering matrix—canceling spurious poles and guaranteeing generalized unitarity—while still being able to close channels below thresholds.

中文翻译：

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R-矩阵理论中复波数的散射矩阵极点展开

在这篇 Ducru等人的后续文章中。[物理。Rev. C 103 , 064608 (2021)]，我们建立了复杂波数的散射矩阵极点扩展的新结果$\mathrm{电阻}$-矩阵理论。过去，出现了两个理论形式的分支来描述核物理中的散射矩阵：$\mathrm{电阻}$-矩阵理论和极点扩展。两人已经相当疏远了。最近，我们对 Brune 的替代参数化的研究$\mathrm{电阻}$- 矩阵理论表明需要扩展散射矩阵（以及底层的 $\mathrm{电阻}$-矩阵运算符）到复数波数。由于转变定义的历史模糊性，出现了两种相互竞争的方法$\mathit{秒}$ 和渗透 $\mathit{磷}$函数：遗留的 Lane 和 Thomas 的“力闭合”方法与解析延拓（这是数学物理学中的标准）。这$\mathrm{电阻}$-matrix 社区尚未就标准核数据库（如 ENDF）中的评估采用哪种方法达成共识。在本文中，我们主张支持分析延续$\mathrm{电阻}$- 矩阵运算符。我们搭桥$\mathrm{电阻}$- 矩阵理论与 Humblet-Rosenfeld 极点扩展，并发现 Siegert-Humblet 放射性极点和宽度的新特性，包括它们对通道半径变化的不变性 ${\mathrm{一种}}_C$. 然后我们证明了$\mathrm{电阻}$-matrix 算子保留了散射矩阵的重要物理和数学特性——消除杂散极点并保证广义单一性——同时仍然能够关闭低于阈值的通道。