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Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions

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.