  We calculate the self-energy of one-dimensional electron band with the
three-dimensional long range Coulomb interaction within the random phase
approximation, paying particular attention to the contribution coming from the
electron scatterings on the collective plasmon mode. It is shown that the
spectral density has a form of wide feature at thr frequency scale of the
plasmon frequency, without the presence of quasi-particle delta-peaks. The
relevance of this result with respect to experimental findings and to the
theory of Luttinger liquids is discussed.
