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

Abstract

The classical Weyl–von Neumann theorem states that for any self-adjoint operator A 0 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 Ext A 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 A ˜ ac and A 0 ac of A ˜ = A ˜ ⁎ ∈ Ext A and A 0 are unitarily equivalent provided that the resolvent difference K A ˜ : = ( A ˜ − 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 + i y ) for a.e. t ∈ R . 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 Ext A .