Simultaneous Strong and Weak Coupling in the Piezoelectric Polaron

When the electron-phonon coupling constant α is much greater than unity, the important phonons have frequencies that are less than the frequencies of the electron-density fluctuations. The strong-coupling (adiabatic) theory then assumes that the lattice acquires a static deformation which traps the electron. However, for acoustic phonons (ω∝k), some of the short-wavelength phonons will generally have high enough frequencies to violate this condition. These phonons should then be treated by perturbation theory. We show that a systematic perturbation theory in α^-1, gives the adiabatic result in lowest order (which is proportional to α²) and a correction (proportional to α) which is the perturbation-theory result for the high-frequency phonons.