Singular perturbations of selfadjoint operators

Singular ̄nite rank perturbations of an unbounded selfadjoint operator A0 in a Hilbert space H0 are de ̄ned formally as A(®) = A0+G®G ¤, where G is an injective linear mapping from H = C to the scale space H¡k(A0), k 2 N, of generalized elements associated with the selfadjoint operator A0, and where ® is a selfadjoint operator inH. 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 ̄nite rank perturbations A(®) in the general setting ranG 1⁄2 H¡k(A0), k 2 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.