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We consider quantum systems with variable but finite number of particles. For such systems we develop geometric and commutator techniques. We use these techniques to find the location of the spectrum, to prove absence of singular continuous spectrum and identify accumulation points of the discrete spectrum. The fact that the total number of particles is bounded allows us to give relatively elementary proofs of these basic results for an important class of many-body systems with non-conserved number of particles.doi:10.1063/1.1452302 fatcat:7wkmxszw7zem7dn2ufg3d5riei