论文标题
两体库仑问题和$ g^{（2）} $代数（再次关于氢原子）
Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom)
论文作者
论文摘要
以氢原子为例，表明，如果三维系统的对称性为$ O（2）\ oplus Z_2 $，则变量$（r，ρ，φ）$允许分离变量$φ$，并且特征函数定义了一个正og的新家族$ new oferthogonal polynomials $ $ $ $ n $ n $ n $ $ n $ n $^railiables^ryalbles^ry^ryphiables^ryphiables^r n ry^ryphiables^ryplybles^re g。这些多项式与代数的有限维表示有关Wolfes模型）。也就是说，这些多项式在研究采菌对氢原子的效应的研究中本质上出现。结果表明，在ic-（r，ρ，φ）中的变量中，在$（r，ρ^2）$的$（r，ρ^2）中的新多项式特征函数中找到了变量。
Taking the Hydrogen atom as an example it is shown that if the symmetry of a three-dimensional system is $O(2) \oplus Z_2$, the variables $(r, ρ, φ)$ allow a separation of the variable $φ$, and the eigenfunctions define a new family of orthogonal polynomials in two variables, $(r, ρ^2)$. These polynomials are related to the finite-dimensional representations of the algebra $gl(2) \ltimes {\it R}^3 \in g^{(2)}$ (discovered by S Lie around 1880 which went almost unnoticed), which occurs as the hidden algebra of the $G_2$ rational integrable system of 3 bodies on the line with 2- and 3-body interactions (the Wolfes model). Namely, those polynomials occur intrinsically in the study of the Zeeman effect on Hydrogen atom. It is shown that in the variables $(r, ρ, φ)$ in the quasi-exactly-solvable, generalized Coulomb problem new polynomial eigenfunctions in $(r, ρ^2)$-variables are found.
