(-0.2,1.7) ;

at (2,2.3) ; (1.8,2.15) – (-0.5,1); (1.8,2.15) – (1,-0.5);

at (-4.5,0.5) ; at (-4.5,0.2) ; (-2.1,0.5) – (-1,0.5);

at (-4.67,-0.5) ; at (-4.55,-0.8) ; (-1.8,-0.5) – (-1,-0.5);

Since near $t_\pm(k)$ the discrete spectrum of the operator ${{-\Delta}}_{\widetilde M}$ corresponds to resonance points $z_{t_\pm}(\lambda) \in \mathcal{M}_{t_\pm}$ with ${\operatorname{Im}}(\lambda) \geq 0$, then a first consequence of Theorem \[t1\] is the following result giving a non cluster phenomena of real or non real eigenvalues near $t_\pm(k)$.

\[t2\] Assume that the potential $M$ satisfies Assumption (A). Then, there is no sequence $(\nu_j)_j$ of non real or real eigenvalues of the operator ${{-\Delta}}_{\widetilde M}$ accumulating at $t_\pm(k)$.

Now, thanks to the parametrizations , the embedded eigenvalues of the operator ${{-\Delta}}_{\widetilde M}$ near $t_\pm(k)$ respectively from the left and the right are the resonances $z_{t_\pm}(\lambda) = t_\pm(k) \mp \lambda^2\sqrt{k}$ with $\lambda \in {{\mathbb{R}}}_+$ sufficiently small. Therefore, as a second consequence of Theorem \[t1\] together with [@af Theorem 9], we have the following:

\[t3\] Assume that the potential $M$ satisfies Assumption (A). Then, for any $r > 0$ small enough, the operator ${{-\Delta}}_{\widetilde M}$ has no embedded eigenvalues in $$\left( t_-(k),t_-(k) + r^2 \right) \cup \left( t_+(k) - r^2,t_+(k) \right).$$ In particular, for $M = 