2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160651A model operator $H$ associated to a system describing four particles in interaction, without conservation of the number of particles, is considered. We describe the essential spectrum of $H$ by the spectrum of the channel operators and prove the Hunziker-van Winter-Zhislin (HWZ) theorem for the operator $H.$ We also give some variational principles for boundaries of the essential spectrum and interior eigenvalues.Mathematical Physics81Q10; 35P20; 47N50text