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 αK≈1, where α 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 K²Σbₙαⁿ, which contains no indication that when αK>1 the ground state is better described by strong coupling. It is suggested that when αK>1 the perturbation series approximates a metastable state of the system.