The cluster expansion for potentials with exponential fall-off
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Continuing the work of a previous paper, the Glimm-Jaffe-Spencer cluster expansion from constructive quantum field theory is adapted to treat quantum statistical mechanical systems of particles interacting by potentials that fall off exponentially at large distance. The HamiltonianH0+V need be stable in the extended sense thatH0+4V+BN≧0 for someB. In this situation, with a mild technical condition on the potentials, the cluster expansion converges and the infinite volume limit of the correlation functions exists, at low enough density. These infinite volume correlation functions cluster exponentially. A natural system included in the present treatment is that of matter with ther−1 potential replaced bye−ar/r. The Hamiltonian is stable, but the system would collapse in the absence of the exclusion principle—the potential is unstable. Therefore this system cannot be handled by the classic work of Ginibre, which requires stable potentials.
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