We develop a microscopic approach to the consistent construction of the kinetic theory of dilute weakly ionized gases of hydrogen-like atoms. The approach is based on the framework of the second quantization method in the presence of bound states of particles and the method of reduced description of relaxation processes. Within the approach we developed the first-order perturbation theory over the weak interaction for a system of kinetic equations for the Wigner distribution functions of free fermions of both kinds and their bound states, the hydrogen-like atoms. It is shown that the conditions of low-temperature approximation, of the gas non-degeneracy and the approximation of weak interaction are realistic and can be met in a wide range of temperatures and the densities of the studied system. We obtain dispersion equations for determining the frequency and wave attenuation coefficients in dilute weakly ionized gas of hydrogen-like atoms near the described equilibrium state. In the two-level atom approximation it is shown that in the system there are longitudinal waves of matter polarization and transverse waves with the behavior characteristic of plasmon polaritons. The expressions for the dependence of the frequency and the Landau damping coefficients on the wave vector for all branches of the oscillations detected, are obtained. Quantitative estimations of the characteristics of the elementary perturbations in the system on an example of a weakly ionized dilute gas of Na-23 atoms are presented. The possibility of using the results of the theory developed to describe the properties of a Bose condensate of photons in dilute weakly ionized gas of hydrogen-like atoms is noted and the directions of its generalizations are discussed.
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