Using as examples the Schroedinger equation and the wave equation we show that homogenization of many linear operators with oscillating coefficients could be done in a frame of the adiabatic approximation based on pseudodifferential operators (functions of noncommuting operators) and the Maslov methods. This approach allows one to reproduce well known homogenization results in the other way, but also take into account so-called dispersion effects leading to a change of structure of original equation. We discuss as example the asymptotic of the solution to the Cauchy problem with localized initial data and rapidly oscillating velocity.
This work was done together with J.Bruening, V.Grushin and S.Sergeev.