On the spectral analysis of many-body systems

We describe the essential spectrum and prove the Mourre estimate for quantum particle systems interacting through k-body forces and creation–annihilation processes which do not preserve the number of particles. For this we compute the “Hamiltonian algebra” of the system, i.e. the C∗-algebra C generated by the Hamiltonians we want to study, and show that, as in the N-body case, it is graded by a semilattice. Hilbert C∗-modules graded by semilattices are involved in the construction of C. For example, if we start with an N-body system whose Hamiltonian algebra is CN and then we add field type couplings between subsystems, then the many-body Hamiltonian algebra C is the imprimitivity algebra of a graded Hilbert CN-module.