We study with the help of Argand diagrams the elastic amplitudes in the scalar Lee model in order to determine the relative importance of inelastic thresholds and the Peierls (P) singularity. An input Nθ resonance, constructed separately by either the choice of the cut-off function or by a CDD pole, leads to non-resonant loops in the Argand diagram of the elastic Vθ amplitude which are associated with the N ∗θ threshold. The P-mechanism works neither in the non-CDD case because the Nθ resonance is necessarily too large, nor in the CDD case because of an explicit cancellation. The low-energy part of the Vθ spectrum, in the absence of input resonances, is characterized by simultaneous Vθ boundstates and anti-resonances. The absence of low-energy Vθ resonances, in the non-CDD case, is connected to the cusp in the elastic Nθ denominator at the Nθ threshold.