On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

摘要

The classical Weyl-von Neumann theorem states that for any self-adjoint operator A0 in a separable Hilbert space H there exists a (non-unique) Hilbert-Schmidt operator C=C~* such that the perturbed operator A_0+C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering the set ExtA of self-adjoint extensions of a given densely defined symmetric operator A in H and some fixed A_0=A_0~*∈_(Ext)_A. We show that the ac-parts ?ac and A_0ac of ?=?*∈_(Ext)_A and A_0 are unitarily equivalent provided that the resolvent difference K_?:=(?-i)~(-1)-(A_0-i)-1 is compact and the Weyl function M(·) of the pair {A,A_0} admits weak boundary limits M(t):=w-lim_(y→+0)M(t+iy) for a.e. t∈R{double-struck}. This result generalizes the classical Kato-Rosenblum theorem. Moreover, it demonstrates that for such pairs {A,A_0} the Weyl-von Neumann theorem is in general not true in the class ExtA.