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

TL;DR

This paper studies the spectral properties of Sturm-Liouville operators with operator potentials, showing unitary equivalences among different self-adjoint realizations and applying results to elliptic differential expressions in half-spaces.

Contribution

It establishes conditions under which various self-adjoint realizations of Sturm-Liouville operators are unitarily equivalent in the absolutely continuous spectrum.

Findings

Dirichlet and Neumann realizations are absolutely continuous and unitarily equivalent.

Self-adjoint realizations with certain spectral conditions are unitarily equivalent to the Dirichlet realization.

Compact resolvent difference implies unitary equivalence of absolutely continuous parts.

Abstract

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.