Stable Self-Interacting Pais-Uhlenbeck Oscillator

TL;DR

This paper investigates the stability and Hamiltonian structure of the self-interacting Pais-Uhlenbeck oscillator, showing that it inherently involves both positive and negative energies but can be stable under certain conditions, making it a viable physical model.

Contribution

It demonstrates the stability regions of the self-interacting Pais-Uhlenbeck oscillator and clarifies the conditions under which positive definite Hamiltonians can be realized.

Findings

Self-interacting Pais-Uhlenbeck oscillator has stable solutions in specific parameter regions.

Modifications like sine interaction and unequal masses expand stability regions.

The oscillator can be a physically acceptable system under certain conditions.

Abstract

It is shown that the interacting Pais-Uhlenbeck oscillator necessarily leads to a description with a Hamiltonian that contains positive and negative energies associated with two oscillators. Descriptions with a positive definite Hamiltonians, considered by some authors, can hold only for a free Pais-Uhlenbeck oscillator. We demonstrate that the solutions of a self-interacting Pais-Uhlenbeck oscillator are stable on islands in the parameter space, as already observed in the literature. If we slightly modify the system, by considering a sine interaction term, and/or by taking unequal masses of the two oscillators, then the system is stable on the continents that extend from zero to infinity in the parameter space. Therefore, the Pais-Uhlenbeck oscillator is quite acceptable physical system.