Two electrons in an external oscillator potential: The hidden algebraic structure.

It is shown that the Coulomb correlation problem for a system of two electrons (two charged particles) in an external oscillator potential possesses a hidden sl2-algebraic structure being one of recently-discovered quasi-exactly-solvable problems. The origin of existing exact solutions to this problem, recently discovered by several authors, is explained. A degeneracy of energies in electron-electron and electron-positron correlation problems is found. It manifests the first appearence of hidden sl2-algebraic structure in atomic physics. PASC: 03.65.Fd and 03.65.Ge On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia E-mail: [email protected] or [email protected] The problem of evaluation of effects of inter-electronic interactions is one of the central problems in atomic physics. The main difficulty comes out from the fact that this problem cannot be solved exactly even in particular cases, while numerical solutions are too complicated to gain a proper intuition. Therefore, it is quite important to find and elaborate situations, where this problem can be modelled in some relevant way, admitting exact, analytic solutions. One of such situations has been described recently in [1, 2]. A system of two electrons in an external harmonic-oscillator potential with an additional linear interaction was studied in the relative coordinate, defined by the Hamiltonian 1 H = −∇1 + ωr 1 −∇2 + ωr 2 + 2β |r1 − r2| + λ|r1 − r2| (1) where r1,2 are the coordinates of the electrons and β = 1. Atomic units h̄ = m = e = 1 are used throughout and an overall factor 1 2 is omitted. It was found that for certain values of oscillator frequency ω and the parameter λ some eigenstate of (1) can be obtained analytically. The main purpose of this note is to show that this feature is nothing but a consequence of the fact that (1) is one of recently-discovered quasi-exactly-solvable Schroedinger operators [4, 5]. It implies an existence of a hidden algebraic structure [6]. Hereafter, we will focus on the case λ = 0. Quasi-exactly-solvable problems are quantum-mechanical problems for which several eigenstates can be found explicitly. They occupy an intermediate place between exactly-solvable (like Coulomb potential, harmonic oscillator etc) and non-solvable. The quasi-exactly-solvable Schroedinger equations appear in two forms: (i) the Hamiltonian with an infinite discrete spectrum with several eigenstates known algebraically and (ii) the Hamiltonian depending on a free parameter, say β, and a certain fixed magnitude of energy corresponds to the ith state of the Hamiltonian at ith value of parameter β (where i = 0, 1, 2, . . . , n) . Those problems are named the firstand the second type, respectively. Surprisingly, exactly-solvable problems 1 In [3] so called pseudo atoms were introduced : quantum system made from atomic ones in which all Coulomb interactions are replaced by oscillator ones, attractive or repulsive as the case may be. The system described by (1) can be treated as modified twoelectron pseudo atom: where Coulomb attractive interactions are replaced by oscillator ones, while Coulomb repulsion remains unchanged or modified by the linear interaction. Precisely speaking, this means that for the parameters β0, β1, . . . , βn , the ground-