A fully interacting many-electron-phonon system in one dimension: an exactly soluble model

A one-dimensional electron gas problem is investigated within the framework of the Luttinger model incorporating both the electron-acoustic-phonon int eraction and the electron-optical-phonon interaction together with the elec tron-electron interaction. It is shown that this model is exactly soluble a nd the energy spectrum is constituted by bosonic excitations. It is also sh own that for certain values of the coupling parameters the system exhibits an instability indicating the possibility of a phase transition.