One particle interacting with the acoustical phonons in a discrete chain

We have applied a variational method to study both the Davydov model and the Su, Schrieffer, and Heeger model for the interaction between an electron and the acoustical phonons of a discrete chain. A class of localized solutions has been found for both of these models. These solutions appear in the intermediate values of the adiabaticity ?~1 when the particle-phonon coupling constant ? is near the critical point for the delocalization-localization transition. We have widely discussed the limits of the continuum approximation, stressing the importance of the application of the method to a discrete system if we want to explore the transition from a large mobile polaron to a strong localized heavy polaron. Furthermore, we have considered the connection between localization and transport in the low-density limit and the possible applications of the method to understanding of the physical properties of some systems with low-dimensionality.