$d$维类氦离子的解析性和广义基态Majorana解
Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions
摘要：
研究了在$d-$维空间$\mathbb{R}^d$（$d>1$）中具有静态核的非相对论类氦离子$(-e,-e,Ze)$.在假定$r^{-1}$库仑相互作用的条件下，采用双参数关联Hylleraas型试探函数计算了体系在$Z \leq 10$区域内的基态能量.对于奇$d=3,5$，变分能量是由一个合理的代数函数的变分参数，而即使$d=2,4$它是第一次显示，它对应于一个更复杂的非代数表达式。这种双重分析性适用于任何$d$。它使我们能够构建合理准确的近似解的基态能量$E_0(Z,d)$的形式紧凑的解析表达式。我们称之为广义Majorana解。它们再现了著名的$\frac{1}{Z}$展开式中的第一个主导项，并作为某些相关性相关性质的生成函数。（第一）临界电荷$Z_{\rm c}$与$d$和香农熵$S_{r}^{(d)}$与$Z$也计算在本变分方法。根据这些结果，对于物理上重要的情况$d=3$一个更一般的3参数相关Hylleraas型试验是用来计算有限质量效应的马约拉纳解决方案的三体库仑系统与任意电荷和质量。它也可以直接推广到任何$d$。系统$e^-\,e^-\,e^+$，$H_2^+$和$H^-$的具体结果明确指出。我们的变分分析结果与文献报道的精确数值非常吻合。
英文摘要：
Non-relativistic Helium-like ions $(-e,-e,Ze)$ with static nucleus in a $d-$dimensional space $\mathbb{R}^d$ ($d>1$) are considered. Assuming $r^{-1}$ Coulomb interactions, a 2-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domain $Z \leq 10$. For odd $d=3,5$, the variational energy is given by a rational algebraic function of the variational parameters whilst for even $d=2,4$ it is shown for the first time that it corresponds to a more complicated non-algebraic expression. This twofold analyticity will hold for any $d$. It allows us to construct reasonably accurate approximate solutions for the ground state energy $E_0(Z,d)$ in the form of compact analytical expressions. We call them generalized Majorana solutions. They reproduce the first leading terms in the celebrated $\frac{1}{Z}$ expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge $Z_{\rm c}$ vs $d$ and the Shannon entropy $S_{r}^{(d)}$ vs $Z$ are also calculated within the present variational approach. In the light of these results, for the physically important case $d=3$ a more general 3-parametric correlated Hylleraas-type trial is used to compute the finite mass effects in the Majorana solution for a three-body Coulomb system with arbitrary charges and masses. It admits a straightforward generalization to any $d$ as well. Concrete results for the systems $e^-\,e^-\,e^+$, $H_2^+$ and $H^-$ are indicated explicitly. Our variational analytical results are in excellent agreement with the exact numerical values reported in the literature.
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