  By the Bethe ansatz method we study the energy dispersion of a particle
interacting by a local interaction with fermions (or hard core bosons) of equal
mass in a one dimensional lattice. We focus on the period of the Bloch
oscillations which turns out to be related to the Fermi wavevector of the Fermi
sea and in particular on how this dispersion emerges as a collective effect in
the thermodynamic limit. We show by symmetry that the dispersion is temperature
independent for a half-filled system. We also discuss the adiabatic coherent
collective response of the particle to an applied field.
