The generalized quantum isotonic oscillator

TL;DR

This paper demonstrates that the generalized quantum isotonic oscillator's Schrödinger equation can be transformed into a confluent Heun equation, providing an efficient numerical solution method and identifying algebraic solutions for specific parameters.

Contribution

It introduces a simple algorithm to solve the generalized quantum isotonic oscillator's Schrödinger equation numerically for any parameters and finds algebraic solutions at particular parameter values.

Findings

Transform of Schrödinger equation into confluent Heun form

Efficient numerical solution algorithm applicable to all parameters

Existence of algebraic quasi-polynomial solutions for specific parameters

Abstract

Recently, it has been proved that a nonlinear quantum oscillator, generalization of the isotonic one, is exactly solvable for certain values of its parameters. Here we show that the Schroedinger equation for such an oscillator can be transformed into a confluent Heun equation. We give a very simple and efficient algorithm to solve it numerically, no matter what the values of the parameters are. Algebraic quasi-polynomial solutions, for particular values of the parameters, are found.