 1170 Mark M. Malamud, Hagen Neidhardt
 SturmLiouville boundary value problems with operator potentials and
unitary equivalence
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May 12, 11

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Abstract. Consider the minimal SturmLiouville operator $A =
A_{
m min}$ generated by the differential expression
$\mathcal{A} := rac{d^2}{dt^2} + T$
in the Hilbert space $L^2(\mathbb{R}_+,\mathcal{H})$ where $T = T^*\ge 0$ in $\mathcal{H}$. We investigate the absolutely continuous parts of different selfadjoint realizations of $\mathcal{A}$. In particular, we show that Dirichlet and Neumann realizations, $A^D$ and $A^N$, are absolutely continuous and unitary equivalent to each other and to the
absolutely continuous part of the Krein realization. Moreover,
if $\inf\sigma_{
m ess}(T) = \inf\sigma(T) \ge 0$, then the
part $\widehat{A}^{ac}E_{\widehat{A}}(\sigma(A^D))$ of any selfadjoint realization $\widehat{A}$ of $\mathcal{A}$ is unitarily equivalent to $A^D$. In addition, we prove that the absolutely continuous part
$\widehat{A}^{ac}$ of any realization $\widehat{A}$ is unitarily
equivalent to $A^D$ provided that the resolvent difference
$(\widehat{A}  i)^{1} (A^D  i)^{1}$ is compact. The abstract
results are applied to elliptic differential expression in the halfspace.
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