  We derive a general set of Poor Man's scaling equations and analyze the
stability of the Luttinger state in a system composed of a finite number N of
one dimensional spinless fermionic chains, coupled through a general two body
interaction. The effect of processes with momentum transfer parallel to the
Fermi surface in destroying massless states is investigated. It will be shown
that there are two processes competing: one in which two electrons exchange
chains and the other in which they jump into a same chain. When periodic
boundary conditions in the transverse direction are taken into account this
competition leads always to massive states (except in hyperplanes of the phase
diagram), a well known example being the generalized sine-Gordon model. If
instead open boundary conditions are taken, massless states are possible but
due to this competition the system is placed near instabilities. We argue that
this kind of analysis has relevance for understanding the instabilities of 2D
fermionic systems.
