operator ${{-\Delta}}_{\widetilde M}$ near $z = t_\pm(k)$ as the poles of the above meromorphic extensions. Notice that this set of resonances contains the eigenvalues of the operator ${{-\Delta}}_{\widetilde M}$ localized near the spectral thresholds $t_\pm(k)$. Otherwise, in the two sheets Riemann surfaces $\mathcal{M}_{t_\pm}$, the resonances will be parametrized by $z_{t_\pm}(\lambda)$ for $\lambda$ sufficiently small for technical reasons. Furthermore, the point $\lambda = 0$ corresponds to the threshold $z = t_\pm(k)$ (cf. Section \[sec5\] for more details). Actually, the resonances verifying $$z_{t_\pm}(\lambda) \in \mathcal{M}_{t_\pm}, \quad {\operatorname{Im}}(\lambda) < 0,$$ live in the non physical plane while those verifying $$z_{t_\pm}(\lambda) \in \mathcal{M}_{t_\pm}, \quad {\operatorname{Im}}(\lambda) \geq 0,$$ coincide with the discrete and the embedded eigenvalues of the operator ${{-\Delta}}_{\widetilde M}$ near $t_\pm$ and are localized in the physical plane. We state Theorem \[t1\] where we establish an absence of resonances of the operator ${{-\Delta}}_{\widetilde M}$ near the spectral thresholds $t_\pm(k)$. In particular, this implies results on the absence of discrete spectrum and embedded eigenvalues near $t_\pm(k)$ (cf. Corollaries \[t2\] and \[t3\]). To prove these results, we first reduce the analysis of resonances near the thresholds $t_\pm(k)$ to that of the noninvertibility of some nonself-adjoint compact operators near $\lambda = 0$ (cf. Propositions \[p5,2\] and 