Mark M. Malamud, Hagen Neidhardt
On the unitary equivalence of 
absolutely continuous parts of self-adjoint extensions
(578K, pdf)

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.