纯倒立四次振荡器的无时序关联子：经典混沌遇上量子稳定性
Out-Of-Time-Ordered-Correlators for the Pure Inverted Quartic Oscillator: Classical Chaos meets Quantum Stability
AI总结：
本文研究了纯倒立四次振荡器的OTOCs，发现其仅表现振荡行为而非指数增长，并通过解析与数值方法证明该系统具有实正能谱和可归一化波函数，揭示了经典不稳定与量子稳定性的差异。
摘要：
无时序关联子（OTOCs）被建议作为诊断量子力学系统混沌行为的手段。最近发现，倒立量子谐振子的OTOCs表现出指数增长，反映了该系统在经典和量子力学上均不稳定的事实。在本研究中，我探讨了量子力学中倒立非谐（纯四次）振荡器的OTOCs，发现尽管系统在经典上具有不稳定性，但OTOCs仅表现出振荡行为。在较高温度下，OTOCs在后期似乎表现出饱和，与 $-2 \langle x^2 \rangle_T \langle p^2 \rangle_T$ 的值一致。我提供了来自谱Zeta函数和WKB方法的解析证据，以及薛定谔方程的直接数值解，证明倒立四次振荡器具有实正的能量本征谱和可归一化的波函数。
英文摘要：
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 Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.
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