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

Abstract: The classical Weyl-von Neumann theorem states that for any selfadjoint operator A in a separable Hilbert space H there exists a (non-unique) Hilbert-Schmidt operator C = C∗ such that the perturbed operator A + C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator A in H and fixing an extension A0 = A ∗ 0. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions à = Ã∗ and A0 are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function M(·) of a pair {A,A0} admits bounded limits M(t) := w-limy→+0 M(t + iy) for a.e. t ∈ R. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.