Quelques resultats rigoureux pour les systemes a petit nombre de corps et leurs applications

Abstract

This Thesis include two sets of distinct works but both of them are concerned with few body quantum systems. The first part is a contribution to two optimized lower bounds for four body systems: the old and the new optimized lower bounds. The new optimized lower bound seems to be better than any other lower bound derived before. The saturability feature (equality between the exact value of the energy of the ground state and the value of the lower bound) is proven numerically for the harmonie interactions. ln the second part of the thesis, we pay attention to two kinds of problems. The first one is concerned with a generalisation of a new integrable one-dimensionnai two-body model of the Calogero type, in any dimension of space, where the exact solutions of the stationary Schrëdinger equation are found explicitly (energy spectrum and normalisables waves functions). We show also that this O-dimensional model (0;;::2) is supersymmetric and belongs to the class of conformai SU(1,1) invariants systems. The second work is involved with the exact resolution of generalisations of three-body Calogero and CalogerMarchioro-Wolfes problems.