  The Landau-Pekar equations describe the dynamics of a strongly coupled
polaron. Here we provide a class of initial data for which the associated
effective Hamiltonian has a uniform spectral gap for all times. For such
initial data, this allows us to extend the results on the adiabatic theorem for
the Landau-Pekar equations and their derivation from the Froehlich model
obtained in [8, 7] to larger times.
