Out-Of-Time-Ordered-Correlators for the Pure Inverted Quartic Oscillator: Classical Chaos meets Quantum Stability

TL;DR

This paper investigates the behavior of out-of-time-ordered-correlators (OTOCs) in the inverted quartic oscillator, revealing oscillatory dynamics despite classical instability, and provides analytical and numerical evidence of its spectral properties.

Contribution

It is the first to analyze OTOCs in the inverted quartic oscillator, showing oscillatory behavior and spectral stability, contrasting with the exponential growth seen in simpler unstable systems.

Findings

OTOCs exhibit oscillatory behavior in the inverted quartic oscillator.

Spectral analysis shows the system has a real, positive energy spectrum.

At high temperatures, OTOCs saturate to a specific value.

Abstract

Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of $-2 \langle x^2 \rangle_T \langle p^2 \rangle_T$ at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schr\"odinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.