Overlapping of nonlinear resonances and the problem of quantum chaos.

The motion of a nonlinearly oscillating particle under the influence of a periodic sequence of short impulses is investigated. We analyze the Schrödinger equation for the universal Hamiltonian. It is shown that the quantum criterion of overlapping of resonances is of the form lambdaK>or=1, where K is the classical coefficient of stochasticity and lambda is the functional defined with the use of Mathieu functions. The area of the maximal values of lambda is determined. The idea about the emerging of quantum chaos due to the adiabatic motion along the curves of Mathieu characteristics at multiple passages through the points of branching is advanced.