is no restriction on the perturbation potential $M$ concerning its self-adjointness or not. The case $k = 1$ includes in particular the case of the Laplacian on $\ell^2({{\mathbb{N}}},{{\mathbb{C}}})$ without any boundary condition at $0$.

As mentioned above, in this article we investigate the resonances (or eigenvalues) distribution for the operator ${{-\Delta}}_M$ near the spectral thresholds $t_\pm(k)$ given by . As this will be observed, the work of Allard and Froese [@af] will play an important role in our analysis (cf. Section \[sec3\] for more details). More precisely, in order to establish a suitable representation of the resolvent associated to the operator $-L + k + 1$, $k \geq 1$ (cf. Theorem \[p4,1\]). Under Assumption (A), the perturbation potential $\widetilde{M}$ satisfies the decay assumption of $M$. So, if we let $\Lambda_n$ to denote the orthogonal projection onto $\bigoplus_{r=0}^n \ell^2(S_r)$, then with the aid of the Schur lemma, it can be shown that $$\label{eq,l}
\Vert \widetilde{M} - \widetilde{M} \Lambda_n \Vert \underset{n \rightarrow 
\infty}{\longrightarrow} 0.$$ Since $\ell^2(S_r)$ is a $k^r$-dimensional space, then implies that the operator $\widetilde{M}$ is the limit in norm of a sequence of finite rank operators. Therefore, $\widetilde{M}$ is a compact operator and in particular it is relatively compact with 