Time-dependent rational extensions of the parametric oscillator: quantum invariants and the factorization method

TL;DR

This paper introduces new time-dependent potentials for the parametric oscillator using quantum invariants and factorization methods, leading to rational extensions that generalize harmonic oscillator solutions.

Contribution

It develops a novel approach to generate time-dependent rational extensions of the parametric oscillator via quantum invariant factorization.

Findings

New families of time-dependent potentials are constructed.

The solutions of the Schrödinger equation are explicitly obtained.

Rational extensions of the harmonic oscillator are recovered as a special case.

Abstract

New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of the parametric oscillator, leading to new families of quantum invariants that are almost-isospectral to the initial one. Then, the respective time-dependent Hamiltonians are constructed, and the solutions of the Schr\"odinger equation are determined from the intertwining relationships and by finding the appropriate time-dependent complex-phases of the Lewis-Riesenfeld approach. To illustrate the results, the set of parameters of the new potentials are fixed such that a family of time-dependent rational extensions of the parametric oscillator is obtained. Moreover, the rational extensions of the harmonic oscillator are recovered in the appropriate limit.