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

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.