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The Xi Operator and its Relation to Krein's Spectral Shift Function | Fritz Gesztesy
; Konstantin A. Makarov
; | Date: |
13 Apr 1999 | Subject: | Spectral Theory | math.SP | 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 $Xi(J+K^*(H_0-lambda-i0)^{-1}K)$ 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 $(E_{J+A(lambda)+tB(lambda)}((-infty,0)),E_J((-infty,0)))$, $tinbR$, 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. | Source: | arXiv, math.SP/9904050 | Services: | Forum | Review | PDF | Favorites |
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