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 $A$ in a separable Hilbert space $\mathfrak H$ 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 $\mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $\widetilde A = {\widetilde A}^*$ and $A_0$ 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 $\{A,A_0\}$ admits bounded limits $M(t) := \wlim_{y\to+0}M(t+iy)$ for a.e. $t \in \mathbb{R}$. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.