Time-dependent pais–uhlenbeck oscillator and its decomposition

Abstract

The Pais–Uhlenbeck(PU) oscillator is the simplest model with higher time derivatives, and its properties has been studied for a long time. In this paper, we extend the 4th-order free PU oscillator to a non-trivial case, dubbed the 4th-order time-dependent PU (tdPU) oscillator, which has timedependent frequencies. We show that this model cannot be decomposed into two harmonic oscillators in contrast to the original PU oscillator by a linear coordinate canonical transformation derived by Smilga. As a result of sustaining canonicality of this transformation for the tdPU oscillator, an interaction is added.