Singularities of the wave trace for the Friedlander model

In a recent preprint, we showed that for the Dirichlet Laplacian $\Delta$ on the unit disk, the wave trace ${Tr}(e^{it\sqrt{\Delta}})$, which has complicated singularities on $2\pi - \epsilon < t < 2\pi$, is, on the interval $2\pi < t < 2\pi + \epsilon$, the restriction to this interval of a $C^\infty$ function on its closure. In this paper we prove the analogue of this somewhat counter-intuitive result for the Friedlander model. The proof for the Friedlander model is simpler and more transparent than in the case of the unit disk.