Rigorous Results for the One-dimensional Fermi Liquid at Zero Temperature

1 In this paper I present some of the most important aspects for a rigorous discussion of analyticity properties of Schwinger functions for the ground state theory of a many-fermion system in one dimension. This proceeding is based on a work performed in Rome by G. Benfatto, G. Gallavotti, B. Scoppola and myself, to appear as a long paper 3] on Communications in Mathematical Physics with the title \Beta function and Schwinger functions for a many-fermions system in one dimension. Anomaly of the Fermi sur-face". Formalism and ideas of 1] and 2] are also at the basis of this work. Readers interested in technical details can nd them in these papers. Here only main ideas and results will be given. 1 The Model We studied a system of spinless fermions in a one-dimensional box of size L with periodic boundary conditions. System density is xed by the chemical potential ; particles have mass m and interact through a pair potential v(x ? y) which we suppose bounded, smooth and short ranged (we call the interaction parameter). The second quantization Hamiltonian for the model, choosing h = 1, is: (1) where, in the l.h.s. of 1, the rst term is the free hamiltonian, or kinetic energy and p F = (2mm) 1=2 is the Fermi impulse. The second term is the