N-body quantum scattering theory in two Hilbert spaces. VI. Compactness conditions

Abstract

It is shown how to implement in a practical way the approximation theory previously developed [J. Funct. Anal. 52, 80 (1983)] for nonrelativistic N-body quantum systems of particles interacting via pair potentials belonging to a certain general class. This is done by constructing the projection operators Π which generate the approximations, and by proving that certain operators Π(J*J−I)Π are Hilbert–Schmidt and that certain other operators VΠE(Δ) are trace class for all finite real intervals Δ. Two types of projections Π are considered. The results for the first type generalize previous results of Combes and Simon for asymptotic channels with only two clusters. The results for the second type provide an alternative approach to N-body scattering and spectral problems which is both practical and theoretically correct. The compactness results are used to prove that the approximate theories are exact theories for approximate Hamiltonians, that the approximate wave operators are asymptotically complete and satisfy the invariance principle, that the kernels of certain N-body equations are compact, and that the Hunziker–van Winter–Zhislin (HVZ) theorem holds for the approximate systems. Furthermore, the approximate Hamiltonians and wave operators converge to the corresponding exact operators in an appropriate limit as the order of the approximation increases.