Phonon-Induced Renormalization and Interaction: An Improvement on   Frohlich Transformation
  Starting from the Flohlich electron-phonon model, which has a history of 50 years, a new unitary transformation is proposed to implement the perturbation treatment. Our main results are: (1)The phonon-induced interaction shows a crossover from the BCS-like potential when the phonon frequency $\omega_p$ is much smaller than the Fermi energy $E_F$ to that of the small polarons when $\omega_p/E_F\ge 1$. (2)The jump of momentum distribution of electron number $n_{\bf k}$ at the Fermi surface goes to zero when the dimensionless coupling constant $\lambda$ increases to the critical value $\lambda_c\ge 1$, which means a possible broken down of the Fermi-liquid description. 
H. Zheng