Towards the rank-one singular perturbations theory of self-adjoint operators

Abstract

The perturbation theory is developed in the case when an arbitrary positive self-adjoint operator is perturbed by the projector on a generalized vector. Similar to the well-known problem −Δ+λδ MS we obtain in general situation explicit representations for singularly perturbed operators their resolvents find the point spectrum and an explicit form of the corresponding eigenvectors. Our approach differs from usual ones and based on the self-adjoint extensions theory of semibounded operators.