respect to the operator $-L + k + 1$. Thus, since the operator $-L + k + 1$ is self-adjoint, then by [@gohs Theorem 2.1, p. 373] we have a disjoint union $$\sigma({{-\Delta}}_{\widetilde M}) = \sigma_{\text{\bf ess}}({{-\Delta}}_{\widetilde M}) \bigsqcup 
\sigma_{\text{\bf disc}}({{-\Delta}}_{\widetilde M}).$$ Moreover, Weyl’s criterion on the invariance of the essential spectrum implies that $$\sigma_{\textup{\bf ess}} ({{-\Delta}}_{\widetilde M}) = \sigma_{\textup{\bf ess}} 
(-L + k + 1) = \left[ t_-(k),t_+(k) \right].$$ However, the (complex) discrete spectrum $\sigma_{\text{\bf disc}}({{-\Delta}}_{\widetilde M})$ generated by the potential $\widetilde{M}$ can only accumulate at the points of $\sigma_{\textup{\bf ess}} ({{-\Delta}}_{\widetilde M})$.

When ${\widetilde M} = {\widetilde M}^\ast$, $\sigma_{\textup{\bf disc}} ({{-\Delta}}_{\widetilde M})$ is just the set of real eigenvalues of ${{-\Delta}}_{\widetilde M}$ respectively from the right and the left of $t_\pm(k)$.

Exploiting the exponential decay of the potential $\widetilde{M}$, we extend (cf. Section \[sec5\]) meromorphically in Banach weighted spaces the resolvent of the operator ${{-\Delta}}_{\widetilde M}$ near $z = t_\pm(k)$, in some two sheets Riemann surfaces $\mathcal{M}_{t_\pm}$ respectively. The first main difficulty to overcome is to establish a good representation of the kernel of the resolvent associated to the operator $-L + k + 1$, $k \ge 1$ (cf. Section \[sec4\]). We thus define the resonances of the 