  In a Luttinger liquid phase of one-dimensional molecular matter the strength
of zero-point motion can be characterized by dimensionless De Boer's number
quantifying the interplay of quantum fluctuations and two-body interactions.
Selecting the latter in the Morse form we show that dissociation of the
Luttinger liquid is a process initiated at the system edge. The latter becomes
unstable against quantum fluctuations at a value of De Boer's number which is
smaller than that of the bulk instability which parallels the classical
phenomenon of surface melting.
