We propose two complementary approaches in understanding coupled electron-phonon systems. The first one is based on extensive numerical studies of the full time evolution of one or two electrons coupled with a lattice. Our previous results clarify the role of coupling, adiabatic parameter, initial electronic configuration, discreteness, and disorder in the problem of polaron formation and demonstrate the existence of novel, long- lived metastable states. We will extend these studies to include bipolaron formation, higher dimensions, application of external field, quantum and thermal fluctuations.
The second approach is based on rigorous mathematical techniques and numerical methods for the many - electron problem in the adiabatic Holstein model with or without magnetic field. They will be applied in the study of bipolaronic - polaronic ground state distributions in any dimension and will be extended to include temperature, orbitar response, anisotropy, and other physical models. The results are expected to have far reaching consequences in the field of transport properties of solids and elucidate a lot of aspects of the mechanism of high critical temperature superconductivity.