  This paper is part of a program by Feldman, Trubowitz and ourselves to
construct a mathematically rigorous version of the BCS mechanism for
superconductivity. In previous papers the renormalization group around the
Fermi surface was defined and analyzed at all orders in perturbation theory.
Non-perturbative results (also called constructive results) correspond to the
resummation of all graphs in perturbation theory and are harder to obtain. In a
previous paper (Helv. Phys. Acta, 65, 679 (1992) we introduced a cluster
expansion for many Fermion systems in two space dimensions based on a so-called
sector or angular decomposition. However this method does not work in three
space dimensions. In this paper a completely different expansion is introduced,
based on an auxiliary scale decomposition and the use of the Hadamard
inequality. We prove that the perturbative expansion for a single scale of the
renormalization group has a convergence radius independent of the scale. This
is a typical result 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.
