MPEJ Volume 1, No.3, 28pp
Received: September 8, 1995, Revised: October 16, 1995, Accepted: October 23, 1995

Jacques Magnen, Vincent Rivasseau
A Single Scale Infinite Volume Expansion
for Three Dimensional Many Fermion Green's Functions

ABSTRACT:  In [FMRT1] we introduced a cluster expansion for many Fermion
systems in two space dimensions based on a so-called sector decomposition.
In this paper a completely different expansion is introduced to treat
the more difficult case of three (or more) space dimensions: it is based
on an auxiliary scale decomposition and the use of the Hadamard inequality.
We prove that the perturbative expansion for a single scale model has a
convergence radius independent of the scale. This is a typical result,
proved in the two dimensional case in [FMRT1], which we cannot obtain
in three dimensions by naive extrapolation of the sector method.
Although we do not treat in this paper the full (multiscale) system, 
we hope this new method to be a significant step towards the rigorous
construction of the BCS theory of superconductivity in three space dimensions.