\[p5,3\]). This can be seen as a Birman-Schwinger principle in a nonself-adjoint context. Afterwards, the reduction made on the problem is reformulated in terms of characteristic values problems (cf. Propositions \[p6,2\] and \[p6,3\]). This allows us to apply powerful results (cf. Section \[sa\]) on the theory of characteristic values of finite meromorphic operator-valued functions to conclude.

Statement of the main results {#secp}
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Let us first fix some notations. If $\lambda \in {{\mathbb{C}}}$, as usual $\vert \lambda \vert <\!\!<1$ means that $\lambda$ is chosen small enough. The set of resonances of the operator ${{-\Delta}}_{\widetilde M}$ near the spectral thresholds $t_\pm(k)$ given by will be respectively denoted by $${\rm Res}_{t_\pm} \, ({{-\Delta}}_{\widetilde M}).$$ We also recall that near $z = t_\pm(k)$, the resonances are defined in some Riemann surfaces $\mathcal{M}_{t_\pm}$ and coincide with $z_{t_\pm}(\lambda)$, $0 < \vert \lambda \vert <\!\!<1 $. More precisely, they are parametrized respectively by $$\label{p1}
z_{t_\pm}(\lambda) := t_\pm(k) \mp \lambda^2\sqrt{k} \in \mathcal{M}_{t_\pm}.$$ Furthermore, the embedded eigenvalues and the discrete spectrum of the operator ${{-\Delta}}_{\widetilde M}$ near $t_\pm(k)$ are the resonances $z_{t_\pm}(\lambda) \in \mathcal{M}_{t_\pm}$ with ${\operatorname{Im}}(\lambda) \geq 0$.

Now, for $0 < r <\!\!<1$, let us introduce the punctured neighborhood of $\lambda = 0$ $$\Omega_r^\ast := \big \lbrace 