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

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

1999-04-13

arxiv.org/abs/math/9904050v2

Mathematics

Spectral Theory

LaTeX, 35 pages, this is an extended version including a discussion of generalized spectral shift functions and extensions of

Scientific paper

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 $\Xi(J+K^*(H_0-\lambda-i0)^{-1}K)$ associated with the operator-valued Herglotz function $J+K^*(H_0-z)^{-1}K$, $\Im(z)>0$ in $\calH$, where $V=KJK^*$ 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 $(E_{J+A(\lambda)+tB(\lambda)}((-\infty,0)),E_J((-\infty,0)))$, $t\in\bbR$, where $A(\lambda)=\Re(K^*(H_0-\lambda-i0)^{-1}K)$, $B(\lambda)=\Im(K^*(H_0-\lambda-i0)^{-1}K)$ 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 $(\Xi(J+K^*(H_0-\lambda-i0)^{-1}K),\Xi(J))$. 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.

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