Solution of the stochastic Liouville equation for a particle in the field of variable scatterers

The quantum-mechanical scattering of a particle on a number of scatterers is considered. It is assumed that the positions of the scatterers in space and times they are switched on are distributed randomly. The solution of the Liouville equation for the density matrix of a particle in the class of nuclear operators is represented in the form of an absolutely convergent series that admits termwise averaging with respect to the positions and switching on times of the scatterers. As a result of partial summation of this averaged series, an analog is obtained of the Dyson equation with “mass operator” in the first order in the density of scatterers. A rigorous estimate is obtained in the form of an inequality for the difference between the exact value of the averaged density matrix of the particle and the solution of the Dyson equation that is obtained. At the same time, the number of scatterers enters the right-hand side of this inequality only through their density.