H -n-perturbations of Self-adjoint Operators and Kreins Resolvent Formula

Supersingular H -n rank one perturbations of an arbitrary positive selfadjoint operator A acting in the Hilbert space H are investigated. The operator corresponding to the formal expression A_(alpha)=A+(alpha)<(phi),.>,(phi),(alpha)(element of)R,(phi)(element of)H-n( A),is determined as a regular operator with pure real spectrum acting in a certain extended Hilbert space H(supset)H. The resolvent of the operator so defined is given by a certain generalization of Kreinʹs resolvent formula. It is proven that the spectral properties of the operator are described by generalized Nevanlinna functions. The results of [24] are extended to the case of arbitrary integer n >= 4.