  At high magnetic field, the semiclassical approximation which underlies the
Ginzburg-Landau (GL) theory of the mixed state of type II superconductors
breaks down. In a quasi-1D superconductor (weakly coupled chains system) with
an {\it open Fermi surface}, a high magnetic field stabilizes a cascade of
superconducting phases which ends in a strong reentrance of the superconducting
phase. The superconducting state evolves from a triangular Abrikosov vortex
lattice in the semiclassical regime towards a Josephson vortex lattice in the
reentrant phase. We study the properties of these superconducting phases from a
microscopic model in the mean-field approximation. The critical temperature is
calculated in the quantum limit approximation (QLA) where only Cooper
logarithmic singularities are retained while less divergent terms are ignored.
The effects of Pauli pair breaking (PPB) and impurity scattering are taken into
account. The Gor'kov equations are solved in the same approximation but
ignoring the PPB effect. We derive the GL expansion of the free energy. We
obtain the specific heat jump at the transition, the sign of the magnetization
and the quasi-particle excitation spectrum. The calculation is extended beyond
the QLA taking into account all the pairing channels and the validity of the
QLA is discussed in detail.
