The Xi Operator and its Relation to Krein's Spectral Shift Function

Original · EN

We explore connections between Krein's spectral shift function ξ(λ,H₀,H) associated with the pair of self-adjoint operators (H₀,H), H=H₀+V in a Hilbert space and the recently introduced concept of a spectral shift operator Ξ(J+K*(H₀-λ-i0)⁻¹K) associated with the operator-valued Herglotz function J+K*(H₀-z)⁻¹K, (z)>0 in, where V=KJK* and J=(V). Our principal results include a new representation for ξ(λ,H₀,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (EJ+A(λ)+tB(λ)((-∞,0)),Eⱼ((-∞,0))), t∈, where A(λ)=(K*(H₀-λ-i0)⁻¹K), B(λ)=(K*(H₀-λ-i0)⁻¹K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A,P) in, where A is bounded and P is an orthogonal projection, we prove that ξ(λ,H₀,H) coincides with the trindex associated with the pair (Ξ(J+K*(H₀-λ-i0)⁻¹K),Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ-operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman-Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.