The Ξ operator and its relation to Krein's spectral shift function

• Published in: Journal d'analyse mathématique (Jerusalem) Vol. 81; no. 1; pp. 139 - 183
• Main Authors: Gesztesy, Fritz, Makarov, Konstantin A.
• Format: Journal Article
• Language: English
• Published: Jerusalem Springer Nature B.V 01-01-2000
• Subjects: Eigenvalues; Operators (mathematics)
• ISSN: 0021-7670; 1565-8538
• DOI: 10.1007/BF02788988

We explore connections between Krein's spectral shift function ζ(λ,H0, H) associated with the pair of self-adjoint operators (H0, H),H=H0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K*(H0−λ−i0)−1K) associated with the operator-valued Herglotz functionJ+K*(H0−z)−1K, Im(z)>0 inH, whereV=KJK* andJ=sgn(V). Our principal results include a new representation for ζ(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))), ℝ, whereA(λ)=Re(K*(H0−λ−i0−1K),B(λ)=Im(K*(H0−λ-i0)−1K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H0, H) coincides with the trindex associated with the pair (Ξ(J+K*(H0−λ−i0)K), 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.