Breakdown of perturbation theory for the deformation potential polaron

By comparing variational forms of weak and strong coupling we show that the crossover point is near $\alpha K\approx 1$, where $\alpha$ is the coupling constant and $K$ is the largest allowed (dimensionless) wave vector. Analysis of the perturbation series gives a leading term of the form $K^2\Sigma b_n{\alpha }^n$, which contains no indication that when $\alpha K>\mathrm{}1\mathrm{}$ the ground state is better described by strong coupling. It is suggested that when $\alpha K>\mathrm{}1\mathrm{}$ the perturbation series approximates a metastable state of the system.