Vibrational excitations of a one-dimensional electron-phonon system in strong coupling

We consider an electron interacting with optical phonons through the deformation potential in one dimension. We calculate the energy spectrum of such a system to one order in 1α² beyond the strong-coupling limit. We calculate the self-energy, effective mass and modified phonon spectrum to this order. The modified phonon spectrum is determined by the solution of a homogeneous linear integral equation. We are able to solve this integral equation in closed form for the odd-parity phonon modes. The result is a frequency spectrum Ωₙ=ω₀[1-4/n(n+3)]1/2, where n=1,3,5, and ω₀ is the unperturbed optical-phonon frequency. For $n=1$, Ω₁=0 and this mode is a translation. For even-parity modes, the phonon frequency spectrum is determined numerically.