Factorization procedure and new generalized Hermite functions

TL;DR

This paper introduces a novel factorization method for the harmonic oscillator Hamiltonian, leading to new generalized Hermite functions and a Sturm-Liouville eigenvalue problem that unifies several classical equations.

Contribution

It presents an alternative factorization approach that generalizes Hermite functions and encompasses known factorizations like Mielnik's as special cases.

Findings

Derivation of a new Sturm-Liouville eigenvalue equation

Introduction of generalized Hermite functions

Unification of Schrödinger and Hermite equations

Abstract

We propose an alternative factorization for the simple harmonic oscillator hamiltonian which includes Mielnik's isospectral factorization as a particular case. This factorization is realized in two non-mutually adjoint operators whose inverse product, in the simplest case, lead to a new Sturm-Liouville eigenvalue equation which includes Schrodinger's equation for the oscillator and Hermite's equation as particular cases for limiting values of the factorization's parameter, and whose eigenfunctions allow us to define new generalized Hermite functions.