A fully solvable quantum theory in which the spectrum of scattering states shifts with the strength of the interaction is presented. Inelastic processes in which two particles go into three are obtained in this cascade model. By using the formalism of analytic continuation of the state spaces, resonances and redundant poles are identified. The completeness of the states is explicitly demonstrated and the eigenchannels when two- and three-particle channels are open are determined. This is one of the few models of scattering of an unstable particle.