 09103 Mark M. Malamud, Hagen Neidhardt
 On the unitary equivalence of
absolutely continuous parts of selfadjoint extensions
(578K, pdf)
Jul 3, 09

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Abstract. The classical Weylvon~Neumann theorem
states that for any selfadjoint operator $A$ in a separable
Hilbert space $\mathfrak H$ there exists a (nonunique) HilbertSchmidt
operator $C = C^*$ such that the perturbed
operator $A+C$ has purely
point spectrum. We are interesting whether this result remains
valid for nonadditive perturbations by considering selfadjoint
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 SturmLiouville operators with operator potentials.
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