On the unitary equivalence of absolutely continuous parts of self-adjoint extensions : dedicated to the memory of M. S. Birman

Abstract

The classical Weyl-von Neumann theorem states that for any self-adjoint operator A in a separable Hilbert space gotH 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 mathfrakH and fixing an extension A0=A0∗. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions widetildeA=widetildeA∗ 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(cdot) of a pair A,A0 admits bounded limits M(t):=wlimyto+0M(t+iy) for a.e. tinmathbbR. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.