Magnen J., Rivasseau V.
A Single Scale Infinite Volume Expansion for Three-Dimensional
Many Fermion Green's Functions
(58K, TeX)
ABSTRACT. 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.
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
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.