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.
Categories: math-ph math.FA math.MP