the following determination of the complex square root $$\label{eq5,1}
{{\mathbb{C}}}\setminus (-\infty,0] \overset{\sqrt{\cdot}}
{\longrightarrow} {{\mathbb{C}}}^+ := \big\lbrace z \in {{\mathbb{C}}}: {\operatorname{Im}}(z) > 0 \big \rbrace$$ will be adopted throughout this paper. For $\varepsilon > 0$ such that $0 < \varepsilon < \frac{\delta}{4}$, we let $D(0,\varepsilon)^\ast$ be the punctured neighborhood of $0$ defined by $$\label{eq5,2}
D(0,\varepsilon)^\ast := \big \lbrace \lambda \in {{\mathbb{C}}}: 0 < \vert \lambda \vert < 
\varepsilon \big \rbrace.$$ Thanks to the first change of variables in , to define and to study the resonances of the operator ${{-\Delta}}_{\widetilde M}$ near the spectral thresholds $t_\pm(k)$, it suffices to define and to study them respectively near $u = 0$ and $u = 4$. However, in practice, there is a simple way (see the comments just after Definition \[d5,1\]) allowing to reduce the analysis of the resonances near the second threshold $t_+(k)$ to that of the first one $t_-(k)$. For further use, let $e_{\pm}$ be the multiplication operators by the functions $$\label{eq5,20}
v \longmapsto e_\pm (v) := e^{\pm\frac{\delta}{2}\vert v \vert}.$$ We have the following lemma:

\[l5,1\] Let $z_{t_-}(\lambda)$ be the parametrization defined by . Then, there exists $0 < \varepsilon_0 \le \frac{\delta}{8}$ small enough such that the operator-valued function $$\label{eq5,3}
\lambda \mapsto 