By introducing an $N\theta$ scattering resonance, the $V\theta$, $N\theta \theta$ sector of the Lee model provides a calculable model for reaction processes leading to a three-particle final state where two overlapping resonances can occur. The $V\theta$ scattering and reaction cross sections calculated are found to be free of any bump in the energy range corresponding to overlap of the two final $N\theta$ resonances. The $V\theta$ scattering amplitude is analytically continued to higher Riemann sheets, and the singularity analogous to that conjectured by Peierls to be the cause of higher resonances in the pion-nucleon system has been shown to lie on a Riemann sheet far from the physical domain. These results cast considerable doubt on the validity of this proposed mechanism for the production of higher resonances.

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