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

الأصل · EN

Consider the minimal Sturm-Liouville operator A = A min generated by the differential expression A:= -d²/dt² + T in the Hilbert space L²(R+,H) where T = T*≥ 0 in H. We investigate the absolutely continuous parts of different self-adjoint realizations of A. In particular, we show that Dirichlet and Neumann realizations, Aᵈ and Aⁿ, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if σess(T) = σ(T) ≥ 0, then the part AacEA(σ(Aᵈ)) of any self-adjoint realization A of A is unitarily equivalent to Aᵈ. In addition, we prove that the absolutely continuous part Aac of any realization A is unitarily equivalent to Aᵈ provided that the resolvent difference (A - i)⁻¹- (Aᵈ - i)⁻¹ is compact. The abstract results are applied to elliptic differential expression in the half-space.