\lambda \in {{\mathbb{C}}}: 0 < \vert \lambda \vert < r \big \rbrace.$$ We then can state our first main result that gives an absence of resonances of the operator ${{-\Delta}}_{\widetilde M}$ near the thresholds $t_\pm(k)$, in small domains of the form $t_\pm(k) \mp \sqrt{k} \, {\Omega_r^\ast}^2$.

\[t1\] Assume that the potential $M$ satisfies Assumption (A). Then, for any $r > 0$ small enough and any punctured neighborhood $\Omega_r^\ast$, we have $$\# \big \lbrace z_{t_\pm}(\lambda) \in Res_{t_\pm}({{-\Delta}}_{\widetilde M}) : \lambda \in 
\Omega_r^\ast \big \rbrace = 0,$$ the resonances being counted with their multiplicity given by and .

Notice that Theorem \[t1\] just says that the operator ${{-\Delta}}_{\widetilde M}$ has no resonances in a punctured neighborhood of $t_\pm(k)$ in the two-sheets Riemann surfaces $\mathcal{M}_{t_\pm}$ where they are defined.

\[fig 1\]

=\[fill=gray!60\] =\[fill=white!100\] =\[fill=gray!20\]

(0,0) – (180:1.58) arc (180:360:1.58) – cycle; (0,0) – (0:1.58) arc (0:180:1.58) – cycle; (0,0) circle (0.5); (-2.5,0) – (2.3,0); (2.3,0) node\[right\] ; (0,-2.5) – (0,2.7); (0.05,2.7) node\[right\] ; (0.26,-0.40) node\[above\] ; (0,0) – (0.24,-0.45); (0,0) – (1.31,0.9); (0.8,0.55) node\[above\] ;

at (-1.4,-1.2) ;

at (0.9,-1.4) ; at (1.8,-0.2) ; at (1.4,-1.2) ;

at (0.9,1.4) ; at (1.4,1.2) ;

at (-1.7,0.4) ; at (-1.4,1.2) ;

at (-0.5,1.7) ;

at 