Correlation Functions of Classical and Quantum Artin System defined on Lobachevsky Plane and Scrambling Time
We consider the quantization of the Artin dynamical system defined on the fundamental region of the Lobachevsky plane. This fundamental region of the modular group has finite volume and infinite extension in the vertical axis that correspond to a cusp. In classical regime the geodesic flow in this fundamental region represents one of the most chaotic dynamical systems, has mixing of all orders, Lebesgue spectrum and non-zero Kolmogorov entropy. The classical correlation functions decay exponentially with an exponent proportional to the entropy. Here we calculated the quantum mechanical correlation functions in the approximation when the oscillation modes in the horizontal direction are neglected. The effective influence of these modes is taken into account to some extent by using the exact expression for the reflection amplitude. By performing a numerical integration we observed that a two-point correlation function decays exponentially and that a four-point function demonstrates tendency to decay with a lower pace. With the numerical data available to us it is impossible to estimate scrambling time t∗ or to confirm its existence in the hyperbolic system considered above, but qualitatively we observe a short time exponential decay of the out-of-time correlation function to almost zero value and then an essential increase with the subsequent large fluctuations.