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

Abstract

We explore connections between Krein's spectral shift function $\xi (\lambda ,H_0,H)$ associated with the pair of self-adjoint operators $(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift operator $\Xi (J+K^{\ast }(H_0-\lambda -i0{)}^{-1}K)$ associated with the operator-valued Herglotz function $J+K^{\ast }(H_0-z{)}^{-1}K$, $\Im (z)>0$ in $\calH$, where $V=KJ{K}^{\ast }$ and $J=\sgn(V)$. Our principal results include a new representation for $\xi (\lambda ,H_0,H)$ in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections $(E_{J+A(\lambda )+tB(\lambda )}((-\infty ,0)),E_J((-\infty ,0)))$, $t\in \mathbb{R}$, where $A(\lambda )=\Re (K^{\ast }(H_0-\lambda -i0{)}^{-1}K)$, $B(\lambda )=\Im (K^{\ast }(H_0-\lambda -i0{)}^{-1}K)$ a.e. Moreover, introducing the new concept of a trindex for a pair of operators $(A,P)$ in $\calH$, where $A$ is bounded and $P$ is an orthogonal projection, we prove that $\xi (\lambda ,H_0,H)$ coincides with the trindex associated with the pair $(\Xi (J+K^{\ast }(H_0-\lambda -i0{)}^{-1}K),\Xi (J))$. In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of $\Xi$-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.

LaTeX, 35 pages, this is an extended version including a discussion of generalized spectral shift functions and extensions of the Birman-Schwinger principle