  Consider the minimal Sturm-Liouville operator $A = A_{\rm min}$ generated by
the differential expression $\mathcal{A} := -\frac{d^2}{dt^2} + T$ 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
self-adjoint 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 $\inf\sigma_{ess}(T) = \inf\sigma(T) \ge
0$, then the part $\widehat{A}^{ac}E_{\widehat{A}(\sigma(A^D))$ of any
self-adjoint 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 half-space.
