Perturbation of an elliptic operator by a narrow potential in an $n$-dimensional domain

We study a discrete spectrum of an elliptic operator of the second order in an $n$-dimensional domain, $n\geq2$, perturbed by a potential depending on two parameters, one of the parameters describes the length of the support of the potential and the inverse of the other corresponds to the magnitude of the potential. We give the relation between these parameters, under which the generalized convergence of the perturbed operator to the unperturbed one holds. Under this relation we construct the asymptotics w.r.t. small parameters of the eigenvalues of the perturbed operators.