Parametric oscillators from factorizations employing a constant-shifted Riccati solution of the harmonic oscillator

TL;DR

This paper explores how shifting the Riccati solution in the factorization of the harmonic oscillator leads to new parametric oscillators, some of which may have physical relevance and exhibit PT symmetry.

Contribution

It introduces a novel approach by applying a constant shift to the Riccati solution in oscillator factorization, revealing new classes of parametric oscillators with potential physical applications.

Findings

Certain shifted Riccati solutions produce strictly periodic and antiperiodic oscillators.

Some resulting oscillators exhibit parity-time (PT) symmetry.

The study provides explicit solutions and parameter values for these oscillators.

Abstract

We determine the kind of parametric oscillators that are generated in the usual factorization procedure of second-order linear differential equations when one introduces a constant shift of the Riccati solution of the classical harmonic oscillator. The mathematical results show that some of these oscillators could be of physical nature. We give the solutions of the obtained second-order differential equations and the values of the shift parameter providing strictly periodic and antiperiodic solutions. We also notice that this simple problem presents parity-time (PT) symmetry. Possible applications are mentioned