The chaos conservation problem in quantum physics

We develop a new method by which an asymptotic expansion in powers of N--1/2 as N --> infinity can be constructed for the solution, depending on N arguments, of the Cauchy problem for a special type of equations. The equations covered by our method include the many-particle Schrodinger, Wigner, and Liouville equations for a system of a large number of particles in which the external potential is O(1) and the coefficient of the particle interaction potential is 1/N; the potentials can be arbitrary smooth bounded functions. We apply this method to the equations for N-particle states corresponding to the Nth tensor power of an abstract Hamiltonian algebra of observables. In particular, for the case of many-particle Schrodinger type equations, we show that the approximate equality of the N-particle wave function for large N to the product of one-particle wave functions is not preserved in temporal evolution, although this property is known to be preserved for the finite-order correlation functions (this quantum analog of the chaos conservation hypothesis put forward by M. Kac in 1956 was proved by the analysis of a BBGKY-like hierarchy of equations). To find the leading asymptotic term of the N-particle wave function, one must combine the solution of the well-known Hartree equation, which can be derived from the BBGKY approach, with the solution of another equation (of Riccati type), presented in this paper. We also consider another interesting case, in which the N-particle system under consideration is supplemented by one more particle that interacts with the system so that the coefficient of the interaction potential is O(1). In this case one encounters a system of Hartree equations rather than one equation, and chaos is not conserved even for the correlation functions.