  An analytic theory for the spectral function for electrons coupled with
phonons is formulated in the adiabatic limit. In the case when the chemical
potential is large and negative $\mu \to -\infty$ the ground state does not
have the adiabatic deformation and the spectral function is defined by the
standard perturbation theory. In this limit we present the spectral function
calculated up to fourth order in the electron-phonon coupling constant which
satisfies the sum rules up to the 5th moment. In the case when the chemical
potential is pinned at the polaron binding energy the spectral function is
defined by the ground state with a nonzero adiabatic deformation. We calculate
the spectral function with the finite polaron density in the adiabatic limit.
We also demonstrate how the sum rules for higher moments may be evaluated in
the adiabatic limit. Contrary to the case of zero polaron density the spectral
function with the finite polaron concentration has some contributions which are
characteristic for polarons.
