Singular Perturbations of Self-Adjoint Operators

Singular finite rank perturbations of an unbounded self-adjoint operator A0 in a Hilbert space h0 are defined formally as A(α) = A0 + GαG*, where G is an injective linear mapping from H = Cd to the scale space h−k(A0), k ∈ N, of generalized elements associated with the selfadjoint operator A0, and where α is a self-adjoint operator in H. The cases k = 1 and k = 2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k = 2n > 1 has been considered recently by various authors from a mathematical point of view. In this paper, singular finite rank perturbations A(α) in the general setting ran G ⊂ h−k(A0), k ∈ N, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application, singular perturbations of the Dirac operator are considered.