An alternative factorization of the quantum harmonic oscillator and two-parameter family of self-adjoint operators

TL;DR

This paper presents a new two-parameter factorization of the quantum harmonic oscillator Hamiltonian, unifying various known oscillators and Hermite operators within a broader self-adjoint operator framework.

Contribution

It introduces an alternative two-parameter factorization method that generalizes and connects different oscillator models and Hermite operators.

Findings

Unified framework for harmonic oscillators and Hermite operators

Derived limits recover standard and Mielnik's oscillators

Discussed a Bernoulli-type factorization different from previous models

Abstract

We introduce an alternative factorization of the Hamiltonian of the quantum harmonic oscillator which leads to a two-parameter self-adjoint operator from which the standard harmonic oscillator, the one-parameter oscillators introduced by Mielnik, and the Hermite operator are obtained in certain limits of the parameters. In addition, a single Bernoulli-type parameter factorization which is different of the one introduced by M. A. Reyes, H. C. Rosu, and M. R. Gutierrez, Phys. Lett. A 375 (2011) 2145 is briefly discussed in the final part of this work