Singular rank one perturbations

Abstract

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 Aβ = A + βτ*τ for τ being a functional defined in a subspace of H. In particular, we apply these results to Hβ = −Δ + V + β|δ⟩⟨δ|, where δ is the singular perturbation given by δ(φ) = ∫Sφ dσ, where S is a suitable hypersurface in Rn. Using the fact that the singular perturbation τ*τ 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 H0 = Δ + V and Hβ, and we give a result about the existence of a pure point spectrum (eigenvalues) of Hβ. We also study the case β goes to infinity.