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

Consider the minimal Sturm–Liouville operator A=AminA=Amin generated by the differential expressionA:=−d2dt2+T in the Hilbert space L2(R+,H)L2(R+,H) where T=T⁎⩾0T=T⁎⩾0 in HH. We investigate the absolutely continuous parts of different self-adjoint realizations of AA. In particular, we show that Dirichlet and Neumann realizations, ADAD and ANAN, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if infσess(T)=infσ(T), then the part A˜acEA˜(σ(AD)) of any self-adjoint realization A˜ of AA is unitarily equivalent to ADAD. In addition, we prove that the absolutely continuous part A˜ac of any realization A˜ is unitarily equivalent to ADAD provided that the resolvent difference (A˜−i)−1−(AD−i)−1 is compact. The abstract results are applied to elliptic differential expressions in the half-space.