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

TL;DR

This paper investigates conditions under which the absolutely continuous parts of self-adjoint extensions of a symmetric operator are unitarily equivalent, extending classical spectral results to a broader class of perturbations.

Contribution

It establishes that for a wide class of symmetric operators, the absolutely continuous parts of extensions are unitarily equivalent if their resolvent difference is compact and the Weyl function has bounded limits.

Findings

Absolutely continuous parts are unitarily equivalent under specified conditions.

Unitary equivalence holds when the Weyl function admits bounded limits almost everywhere.

Results apply to direct sums and Sturm-Liouville operators with operator potentials.

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.