Non-Hermitian Hydrogen atom

We have constructed a set of non-Hermitian operators that satisfy the commutation relations of the SO(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(3)$$\end{document}-Lie algebra. Using these set of operators we have constructed a non-Hermitian Hamiltonian corresponding to the Hydrogen atom that includes a complex term but with the same spectra as in the Hermitian case. It is also found a non-Hermitian Runge–Lenz vector that represents a conserved quantity. In this way, we obtain a set of non-Hermitian operators that satisfy the commutation relations of the SO(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SO(4)$$\end{document}-Lie algebra.