Time asymptotics of the velocity autocorrelation function in the Lorentz model

For a Lorentz gas of hard spheres a functional formalism is introduced that makes it possible to represent the solution of the BBGKY hierarchy with arbitrary initial conditions in terms of a Green's operator which gives the solution under special conditions, namely, the absence of correlations at the initial time between the test particle and scatterers. It is shown that under some assumptions that are natural from the physical point of view the Laplace transform of the Green's operator and the corresponding mass operator has on the real negative half-axis in the plane of the Laplace variable $z$ a discontinuity with asymptotic behavior $|x|^{3/2}$, $x=\operatorname{Re}z$. Exact expressions are obtained for the operator coefficients in the asymptotic behavior, these generalizing the results of the ring approximation.