Energy-versus-Momentum Relation for the Piezoelectric Polaron

We note that the intermediate-coupling theory for the piezoelectric polaron gives an energy-momentum relation that is quadratic for small $P$ and asymptotes to a straight line with slope equal to the speed of sound at high $P$. We present arguments that indicate that this is the correct qualitative behavior for the piezoelectric polaron. We show, however, that if we modify the Hamiltonian so as to cut off the interaction with the long-wavelength phonons, there is a maximum momentum for which this type of solution exists.