We consider a quantum many-body model describing a system of
electrons interacting with themselves and hopping from one ion to another of a
one dimensional lattice. We show that the ground state energy of such system,
as a functional of the ionic configurations, has local minima in correspondence
of configurations described by smooth $\frac{\pi}{pF}$
periodic functions, if the interaction
is repulsive and large enough and pF is the Fermi momentum of the electrons.
This means physically that a $d=1$ metal develop a periodic distortion of
its reticular structure (Peierls instability). The minima are found solving the
Eulero-Lagrange equations of the energy by a contraction method.