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

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 A0+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 . We show that the ac-parts and of and A0 are unitarily equivalent provided that the resolvent difference is compact and the Weyl function M(⋅) of the pair {A,A0} admits weak boundary limits M(t):=w-limy→+0M(t+iy) for a.e. t∈R. This result generalizes the classical Kato–Rosenblum theorem. Moreover, it demonstrates that for such pairs {A,A0} the Weyl–von Neumann theorem is in general not true in the class ExtA.