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

Abstract

Consider the minimal Sturm-Liouville operator A = A_rm min generated by the differential expression A := -fracd^2dt^2 + T in the Hilbert space L^2(R_+,cH) where T = T^*ge 0 in cH. We investigate the absolutely continuous parts of different self-adjoint realizations of cA. 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 infsigma_ess(T) = infgs(T) ge 0, then the part wt A^acE_wt A(gs(A^D)) of any self-adjoint realization wt A of cA is unitarily equivalent to A^D. In addition, we prove that the absolutely continuous part wt A^ac of any realization wt A is unitarily equivalent to A^D provided that the resolvent difference (wt A