  One--particle interchain hopping in a system of coupled Luttinger liquids is
investigated by use of exact diagonalizations techniques. Firstly, the two
chains problem of spinless fermions is studied in order to see the behaviour of
the band splitting as a function of the exponent $\alpha$ which characterizes
the $1D$ Luttinger liquid. Moderate intra-chain interactions can lead to a
strong reduction of this splitting. The on-set of the confinement within the
individual chains (defined by a vanishing splitting) seems to be governed by
$\alpha$. We give numerical evidence that inter-chain coherent hopping can be
totally suppressed for $\alpha\sim 0.4$ or even smaller $\alpha$ values. The
transverse conductivty is shown to exhibit a strong incoherent part. Even when
coherent inter-chain hopping is believed to occur (at small $\alpha$ values),
it is shown that the coherent Drude weight is always significantly smaller than
the incoherent weight. Implications for the optical experiments in quasi-1D
organic or high-$T_c$ superconductors is outlined.

Condensed Matter