Abstract
Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with 2〈x2〉T〈p2〉T at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.
Highlights
Where the subscript T denotes calculation of the expectation value in a heat bath of temperature T
The spectral form factor decreases at early times, bounces back and reaches a plateau with strong fluctuations
I studied the quantum-mechanical out-of-time-ordered-correlators for quartic interaction potential
Summary
Which has a discrete solution spectrum that is spanned by the associated generalized Laguerre polynomials. After using the steps detailed in appendix A, the energy eigenvalues En and Laguerre coefficients ci are found to be related to the eigenvalues and eigenvectors of Aij, Bij given in eqs. A full solution for the quartic oscillator requires finding the eigenvalues and eigenvectors for K → ∞, for which A, B become infinite matrices. Solving the eigenproblem (3.8) for fixed and finite K, leads to K parity-even and K parity-odd wave-functions ψn(x) that each are exact solutions to (3.2) at 2K points x = ±x1, ±x2 .
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