  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.
