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 A0 = A ∗0 ∈ ExtA. We show that the ac-parts Aac and Aac 0 of A = A ∗ ∈ ExtA and A0 are unitarily equivalent provided that the resolvent difference K A := ( A−i) −1 −(A0 −i) −1 is compact and the Weyl function M(·) of the pair {A,A0} admits weak boundary limits M(t) := w-limy→+0 M(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. © 2010 Elsevier Inc. All rights reserved.