Simultaneous Strong and Weak Coupling in the Piezoelectric Polaron

When the electron-phonon coupling constant $\alpha$ 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 ($\omega \propto k$), 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 ${\alpha }^{-1\mathrm{}}$, gives the adiabatic result in lowest order (which is proportional to ${\alpha }^2$) and a correction (proportional to $\alpha$) which is the perturbation-theory result for the high-frequency phonons.