Taking the Hydrogen atom as an example it is shown that if the symmetry of a three-dimensional system is O(2)⊕Z2, 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,ρ2). These polynomials are related to the finite-dimensional representations of the algebra gl(2)⋉R3∈g(2) (discovered by S Lie around 1880 which went almost unnoticed), which occurs as the hidden algebra of the G2 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,ρ2)-variables are found.
•Prefactor in m-th Coulomb orbital can be written as polynomial in (r,ρ) variables.•Known generalized (QES) radial Coulomb problem admits polynomial eigenfunctions in two (r,ρ) variables.•Hidden algebra g(2) in (generalized, QES) Hydrogen atom is discovered.
Abstract We study the existence of exact solutions either for linear systems of ODEs or for scalar second order linear differential equations with Dirichlet boundary value conditions.
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