Energy spectrum of the optical polaron at finite total momentum

Abstract

In the following discussion we are concerned with the standard Fr\"ohlich model for an optical polaron. We clarify the qualitative properties of the energy spectrum for arbitrary total momentum Q. Concerning the ground-state energy, we establish an effective lower bound. Until now, we have to assume that the electron-phonon coupling parameter $\ensuremath{\alpha}$ does not exceed a specified positive value. Using this bound, we demonstrate that the ground-state energy coincides with the continuum edge for $|Q|>~|{Q}_{C}|,$ ${Q}_{C}$ being finite. Consequently, it is only for $|Q|<|{Q}_{C}|$ that an isolated ground state exists at all. This behavior is strikingly different from that of the corresponding system in lower dimensions, which has been analyzed previously by other authors, the discussion of the three-dimensional case remaining incomplete. Concerning the overall behavior of the ground-state energy as a function of Q and $\ensuremath{\alpha},$ we find an increase (strict decrease) with increasing $|Q|(\ensuremath{\alpha}).$ In addition, we present an approach to the excited states. Interestingly enough, this can be based entirely on the knowledge of the ground-state energy and ground-state wave function.