Let $u \in {{\mathbb{C}}}\setminus [0,4]$ and introduce the following change of variables $$u = 4 \sin^2 \left( \frac{\Phi}{2} \right), \quad {\operatorname{Im}}(\Phi) > 0.$$ Then, it can be proved (cf. e.g. [@it Section 2]) that we have $$\Big( \mathcal{F}^{-1} \big( 2 - 2 \cos (\cdot) - u \big)^{-1} \Big)(n)
= \frac{ie^{i \vert n \vert \Phi}}{2 \sin(\Phi)} = 
\frac{ie^{i \vert n \vert \Phi}}{\sqrt{u} \sqrt{4 - u}},$$ or equivalently $$\label{eq4,16}
\frac{1}{\pi} \int_{{{\mathbb{T}}}} \big( 2 - 2 \cos(\theta) - u \big)^{-1} e^{in\theta} \, d\theta
= \sqrt{\frac{2}{\pi}} \frac{ie^{i \vert n \vert \Phi}}{\sqrt{u} \sqrt{4 - u}}.$$ Now, with the help of the transformations $\sin(n\theta) = \frac{1}{2i} \left( e^{in\theta} - e^{-in\theta} \right)$ and $$\Big( -2\sqrt{k} \cos(\theta) + k + 1 - z \Big)^{-1} = \frac{1}{\sqrt{k}}
\Big( 2 - 2 \cos(\theta) - u \Big)^{-1},$$ where $u = \frac{z + 2\sqrt{k} - (k + 1)}{\sqrt{k}}$ give immediately . This completes the proof of the theorem.

Resonances near $z = t_\pm(k)$ {#sec5}
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Definition of the resonances
----------------------------

In this subsection, we define the resonances of the operator ${{-\Delta}}_{\widetilde M}$ near the spectral thresholds $t_\pm(k)$ given by . As preparation, preliminary lemmas will be proved firstly.

From now on, the potential perturbation $M$ is assumed to satisfy Assumption (A). Moreover, 