Singular Rank One Perturbations of Self-Adjoint Operators and Krein Theory of Self-Adjoint Extensions

摘要

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A_(infinity) #omega# for A_(#alpha#, #omega#)=A + #alpha#(centre dot, #omega#)#omega#, #alpha#->infinity where A is a self-adjoint positive operator, #omega# implied by H_(-1) (H_s, s implied by R~1 being the 'A-scale'). In the present note it is remarked that the operator A_(infinity,#omega#) also appears directly as the Friedrichs extension of the symmetric operator A :=A {f implied by D (A) | =0}. It is also shown that Krein's resolvents formula: (A_(b,#omega#)-z ~(-1)) = (A-z)~(-1) + b_z~(-1) (centre dot, #eta#z-bar)#eta#z, with b_z = b - (1 + z)(#eta#z, #eta#_(-1)), #eta#_z = (A - z)~(-1)#omega# defines a self-adjoint operator A_(b,#omega#) for each #omega# implied by H_(-2) and b implied by R~1. Moreover it is proven that for any sequence #omega#_n implied by H_(-1) which goes to #omega# in H_(-2) there exists a sequence #alpha#_n->0 such that A_(#alpha#_n,#omega#_n)-> A_(b,#omega#) in the strong resolvent sence.