Sextic anharmonic oscillators and orthogonal polynomials

TL;DR

This paper explores the relationship between sextic anharmonic oscillators and orthogonal polynomials, revealing their properties and connections to quasi-exact solvability and previously discovered polynomial classes.

Contribution

It demonstrates that wave functions of sextic oscillators serve as generating functions for orthogonal polynomials and links them to known polynomial classes through scaling transformations.

Findings

Wave functions generate orthogonal polynomials in energy.

Properties of these polynomials are analyzed in detail.

Connection to Bender-Dunne polynomials is established.

Abstract

Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced,by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.