A weakly-onedimensional heterostructure is a quantum wire structure tailored so that the carriers are tightly confined in a direction, weakly confined in another and free to move in the third. The electronic bound states are computed using the envelope function approximation in the frame on the k.p. theory and the adiabatic approximation. Transition rates of the electrons from ally bound stale to ally other bound state mediated bg the LO-phonons are then evaluated. The lifetimes of excited levels and average energy with respect to time are computed.