Singular Perturbations of Self-Adjoint Operators

Abstract

Singular finite rank perturbations of an unbounded self-adjoint operator A 0 in a Hilbert space ℌ0 are defined formally as A (α)=A 0+GαG *, where G is an injective linear mapping from ℋ=ℂ d to the scale space ℌ-k(A0)k ∈ ℕ, k∈N, of generalized elements associated with the self-adjoint operator A 0, 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 (A 0), 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.