Generalized Rayleigh-Schroedinger perturbation theory as a method of linearization of the so called quasi-exactly solvable models

TL;DR

This paper introduces a linearization method for quasi-exactly solvable models, specifically the sextic oscillator, enabling simultaneous exact solutions for all states with explicit matrix formulas, simplifying calculations and removing degeneracies.

Contribution

It presents a novel N-independent linear perturbation approach for QES models, providing explicit matrix solutions and improving upon traditional nonlinear algebraic methods.

Findings

Exact solvability for all QES states in leading orders

Explicit matrix formulas for perturbation corrections

Degeneracy removal and equidistant spectrum in unperturbed case

Abstract

Sextic oscillator in D dimensions is considered as a typical quasi-exactly solvable (QES) model. Usually, its QES N-plets of bound states have to be computed using the coupled Magyari's nonlinear algebraic equations. We propose and describe an alternative linear method which is N-independent and works with power series in 1/\sqrt(D). Main merit: simultaneous exact solvability (for all the QES states) in the first two leading orders (the degeneracy is completely removed, the unperturbed spectrum is equidistant). An additional merit: All the perturbation corrections are given by explicit matrix formulae in integer arithmetics (there are no rounding errors).