On approximation of high order singular perturbations

Abstract

Let L be a non-negative self-adjoint operator in a Hilbert space H0 with inner product �· , ·� 0 and let ϕ be a singular element belonging to H−k−1\H−k with k 2 (high order), where {Hs} ∞=−∞ is the scale of Hilbert spaces associated with L in H. For the formal singular perturbation of L generated by ϕ self-adjoint realizations H( g) in a Pontryagin space are considered and approximations of these realizations by suitable smoother models are investigated. The realizations H( g)are described by k real parameters (gs) k=1 and given in a Pontryagin space �( ¯ g) of the form H0 ⊕ C m ⊕ C m , which is