Time dependent Pais-Uhlenbeck oscillator and its decomposition

TL;DR

This paper extends the Pais-Uhlenbeck oscillator to include time-dependent frequencies, demonstrating it cannot be decomposed into harmonic oscillators and exploring the effects of coordinate transformations.

Contribution

It introduces a time-dependent Pais-Uhlenbeck oscillator and shows it cannot be decomposed into harmonic oscillators, unlike the original model.

Findings

The time-dependent PU oscillator cannot be decomposed into harmonic oscillators.

Coordinate transformation by Smilga introduces interaction effects.

Extended model exhibits non-trivial dynamics due to time dependence.

Abstract

The Pais-Uhlenbeck(PU) oscillator is the simplest model with higher time derivatives. Its properties were studied for a long time. In this paper, we extend the 4th order free PU oscillator to a more non-trivial case, dubbed the 4th order time dependent PU oscillator, which has time dependent frequencies. We show that this model cannot be decomposed into two harmonic oscillators in contrast to the original PU oscillator. An interaction is added by the coordinate transformation of Smilga.