A quantum exactly solvable non-linear oscillator related with the isotonic oscillator

TL;DR

This paper introduces a new exactly solvable quantum oscillator potential that interpolates between the harmonic and isotonic oscillators, providing explicit wave functions, energies, and a related family of orthogonal polynomials.

Contribution

It presents a novel nonpolynomial quantum potential, proves its exact solvability, and derives associated orthogonal polynomials related to Hermite polynomials.

Findings

Explicit solutions for wave functions and energies of the potential.

Identification of a new family of orthogonal polynomials related to Hermite polynomials.

Demonstration of the potential's relation to well-known oscillators.

Abstract

A nonpolynomial one-dimensional quantum potential representing an oscillator, that can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator (harmonic oscillator with a centripetal barrier), is studied. First the general case, that depends of a parameter $a$, is considered and then a particular case is studied with great detail. It is proven that it is Schr\"odinger solvable and then the wave functions $\Psi_n$ and the energies $E_n$ of the bound states are explicitly obtained. Finally it is proven that the solutions determine a family of orthogonal polynomials ${\cal P}_n(x)$ related with the Hermite polynomials and such that: (i) Every ${\cal P}_n$ is a linear combination of three Hermite polynomials, and (ii) They are orthogonal with respect to a new measure obtained by modifying the classic Hermite measure.