On a method of solution of the initial value problem for the linearized Boltzmann equation

A study is made of the initial value problem for the linearlzed Boltzmann equation. Its approximate solution in a Fourier representation with respect to the spatial variables is obtained as an expansion in the eigenfunctions of the collision operator perturbed by a translation term. This is an asymptotic expansion in the limit $k\to 0$ and the wave number $k$ in dimensionless variables is of the same order of magnitude as the Knudsen number. Perturbation theory is used to find the second-order corrections to the eigenfunctlons and the thirdorder corrections to the eigenvalues for the case of Maxwell molecules. The method used is compared with the other methods of the kinetic theory of gases.