A generalized quantum nonlinear oscillator

TL;DR

This paper explores various generalizations of the quantum nonlinear oscillator, including exactly solvable, quasi-exactly solvable, and non-Hermitian variants, highlighting shape invariance and solutions via classical orthogonal polynomials.

Contribution

It introduces new exactly solvable potentials with shape invariance and extends the understanding of quantum nonlinear oscillators to non-Hermitian cases.

Findings

Identified shape invariance in generalized potentials

Derived solutions using classical orthogonal polynomials

Extended the class of solvable quantum nonlinear oscillators

Abstract

We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the new exactly solvable potentials possess shape invariance symmetry. The solutions are obtained in terms of classical orthogonal polynomials.