Quasi-exactly solvable polynomial extensions of the quantum harmonic oscillator

Christiane Quesne

doi:10.1088/1742-6596/1071/1/012016

Abstract

The (analytic) sextic oscillator is often considered as the prototype of quasi-exactly solvable (QES) Schrödinger equations, i.e., those Schrödinger equations for which at some ad hoc couplings a finite number of eigenstates can be found explicitly by algebraic means, while the remaining ones remain unknown. Recently, a (non-analytic) QES symmetrized quartic oscillator was introduced and shown to complete the list of (analytic) QES anharmonic oscillators, which does not contain any quartic one. Here we prove that such a quartic oscillator is amenable to an sl(2, ℝ) description. Furthermore, we generalize it to a symmetrized sextic oscillator. The latter is obtained by parting the real line into two subintervals ℝ− and ℝ+ on which the corresponding Schrödinger equations are solved by using the functional Bethe ansatz method, and the resulting wavefunctions and their first derivatives are matched at x = 0. Two categories of QES potentials are obtained: the first one containing the well-known analytic sextic potentials as a subset, and the second one of novel potentials with no counterpart in such a class.

Highlights

The sextic oscillator is often considered as the prototype of quasi-exactly solvable (QES) Schrodinger equations, i.e., those Schrodinger equations for which at some ad hoc couplings a finite number of eigenstates can be found explicitly by algebraic means, while the remaining ones remain unknown
We prove that such a quartic oscillator is amenable to an sl(2, R) description. We generalize it to a symmetrized sextic oscillator. The latter is obtained by parting the real line into two subintervals R− and R+ on which the corresponding Schrodinger equations are solved by using the functional Bethe ansatz method, and the resulting wavefunctions and their first derivatives are matched at x = 0
Exact solutions of the Schrodinger equation are known to be very useful for approximating solutions of more realistic equations appearing in practical problems, because they may give a hint for suggesting a good starting point in perturbation theory or variational calculus

Summary

Introduction

Exact solutions of the Schrodinger equation are known to be very useful for approximating solutions of more realistic equations appearing in practical problems, because they may give a hint for suggesting a good starting point in perturbation theory or variational calculus. The latter is obtained by parting the real line into two subintervals R− and R+ on which the corresponding Schrodinger equations are solved by using the functional Bethe ansatz method, and the resulting wavefunctions and their first derivatives are matched at x = 0. More complicated ones, related to polynomial solutions of generalized Heun equations, can be solved by the recursion relation method [16] or by the functional Bethe ansatz method [17, 18].

Results

Conclusion