Sturm-Liouville boundary value problems with operator potentials and unitary equivalence

Abstract

Consider the minimal Sturm-Liouville operator $A=A_{\min }$ generated by the differential expression $\mathcal{A}:=-\frac{d^2}{d{t}^2}+T$ in the Hilbert space $L^2({\mathbb{R}}_{+},\mathcal{H})$ where $T=T^{\ast }\ge 0$ 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 $\widehat{A}^{ac}E_{\widehat{A}(σ(A^D))$ of any self-adjoint realization $\hat{A}$ of $\mathcal{A}$ is unitarily equivalent to $A^D$. In addition, we prove that the absolutely continuous part ${\hat{A}}^{ac}$ of any realization $\hat{A}$ is unitarily equivalent to $A^D$ provided that the resolvent difference $(\hat{A}-i{)}^{-1}-(A^D-i{)}^{-1}$ is compact. The abstract results are applied to elliptic differential expression in the half-space.