Generalized Rayleigh–Schrödinger Perturbation Theory as a Method of Linearization of the so Called Quasi-Exactly Solvable Models

enters many phenomenological and methodical considerations as a “next-to-solvable” model [1]. In fact, among all the real polynomial interactions, only the harmonic and sextic models can generate an arbitrary N–plet of bound state wavefunctions in an elementary form. All the similar models are often called quasi-exactly solvable (QES, cf. [2]). Unfortunately, the close parallel between the sextic and harmonic oscillator is not too robust and breaks down in practical applications [3]. For example, the Rayleigh–Schrödinger unperturbed propagator ceases to be diagonal in the sextic case [4]. Moreover, the key weakness of any QES model lies in the nonlinearity of its secular equation which has the polynomial form of degree N [5]. Non-numerical determination of the sextic energies is only feasible at N ≤ 4. Otherwise, in a sharp contrast to harmonic case, the values of energies En are only available up to some rounding errors. In order to refresh the parallels we shall describe a new approach to the sextic QES bound state problem. It is based on some surprising results of the symbolic manipulation experiments. They were performed in MAPLE using the technique of Groebner bases. We revealed that the QES energies become equidistant and proportional to integers in the limit of the large spatial dimensions D → ∞. This feature is presented in Sections 2 and 3. In the second step of our analysis one discovers that the systematic evaluation of the Rayleigh– Schrödinger corrections proves feasible in closed form. In spite of the non-diagonality of propagators, a merely slightly modified form of construction can be used. It gives the energy formula E(λ) = E + λE + λE + · · ·+ λ E +O (λK+1) , λ = 1/D.