  Electron backscattering in a Luttinger liquid with an impurity is
investigated in the presence of zero point motion of the phonon lattice. The
impurity can mean either a mass defect, an elastic defect or a pinning defect.
The phonon spectrum is then affected by the presence of the defect, which
enters in the renormalization group equations for backscattering. The RG
equation becomes dependent on the energy cutoff for finite Debye frequency
$\omega_D$ giving rise to finite energy effects. We compute the local density
of states and show how the renormalization group flow is affected by a finite
$\omega_D$.
