Singular rank one perturbations

In this paper, A = B + V represents a self-adjoint operator acting on a Hilbert space H. We set a general theoretical framework and obtain several results for singular perturbations of A of the type A(beta) = A + beta tau*tau for tau being a functional defined in a subspace of H. In particular, we apply these results to H-beta = -Delta + V + beta|delta ><delta|, where delta is the singular perturbation given by delta(phi) = integral(S)phi d sigma, where S is a suitable hypersurface in R-n. Using the fact that the singular perturbation tau*tau is a sort of rank one perturbation of the operator A, it is possible to prove the invariance of the essential spectrum of A under these singular perturbations. The main idea is to apply an adequate Krein's formula in this singular framework. As an additional result, we found the corresponding relationship between the Green's functions associated with the operators H-0 = Delta + V and H-beta, and we give a result about the existence of a pure point spectrum (eigenvalues) of H-beta. We also study the case beta goes to infinity.