类氦离子$(Z,e,e)$的拉格朗日网格方法：对高精度能谱随核电荷数$Z$的插值
Helium-like ions $(Z,e,e)$ in Lagrange Mesh method, interpolating the highly-accurate energy spectra vs. Z
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中文总结 AI 辅助
本文提出类氦离子研究的两种方法：基于拉格朗日网格法的高精度数值方法，以及匹配两类展开的少参数插值公式，可对类氦序列任意激发态能谱实现10-14位有效数字的精度，获多项首次结果。
中文摘要 AI 辅助
本文提出非相对论量子力学中研究类氦原子离子的两种替代方法：（I）一种基于拉格朗日网格法的数值方法，该方法在单处理器模式下，以适度的CPU时间即可对任意核电荷数$Z$的能谱轻松达到14-15位有效数字；（II）一种随$Z$变化的高精度少参数能量插值公式。本文提出的通用插值公式可应用于类氦序列任意激发态的能量，其构建基于匹配大$Z$下的$1/Z$展开，以及围绕F和D Stillinger（1969、1974年）提出的所谓第二临界电荷$Z_B$的整数和半整数次幂的Puiseux展开——该电荷于2019年被本文作者针对基态$1^1S$证实，本文在此对其进行重新研究并扩展到激发态。例如，针对前两个自旋单态$1^1S$、$2^1S$和前两个自旋三重态$2^3S$、$3^3S$，该含9个自由参数的插值公式可对任意物理相关的核电荷数$Z$的能量达到10-14位有效数字的精度，在大$Z$下具有绝对精度，且获得了诸多首次报道的结果。
英文摘要
Two alternative approaches for studying Helium-like atomic ions in non-relativistic quantum mechanics are proposed: (I) a numerical approach, based on the Lagrange-mesh method which can easily reach up to 14-15 significant digits in the energy spectrum for any nuclear charge $Z$ with modest CPU time in single processor mode and (II) a highly-accurate, few-parametric interpolation formula for the energies {\it vs.} $Z$. The interpolation formula of general nature is proposed, it can be applied to the energies of any excited state of the helium-like sequence. It is based on matching the $1/Z$-expansion at large $Z$ and the Puiseux expansion with integer and half-integer powers around the so-called second critical charge $Z_B$, introduced by F and D Stillinger (1969, 1974), confirmed by the present authors in 2019 for the ground state $1^1 S$, then revisited here, and extended to the excited states in the present work. For example for the first two spin-singlet $1^1 S$, $2^1 S$ and the first two spin-triplet $2^3 S$, $3^3 S$ states this interpolation formula with nine free parameters can reach an accuracy of 10-14 significant digits (s.d.) in the energies for any physically-relevant nuclear charge $Z$, giving absolute accuracy at large $Z$. Many results are obtained for the first time.
