The quantum chaos problem

This review describes the present state of the quantum chaos problem: The problem of determining the specific properties of autonomous and nonautonomous quantum systems with few degrees of freedom whose classical analogs have an unstable (stochastic) motion and also that of determining the relationships between these properties and the characteristics of the classical stochastic situation. The criteria for quantum chaos which have been established to date are examined and compared for autonomous systems. These criteria make use of properties of the energy spectrum, the wave functions in various representations, the matrices of operators other than the Hamiltonian, and particular features of the evolution of time-varying states (wave packets) in such systems. For nonautonomous systems, the conditions for the applicability of classical dynamics for describing observables and certain specific quantum-mechanical effects (tunneling through a separatrix and a global quantum resonance) are analyzed.