Toward the rank-one singular perturbation theory of self-adjoint operators

Abstract

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