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Abstract

Given a self-adjoint operator $H_0,$ a self-adjoint trace-class operator $V$ and a fixed Hilbert–Schmidt operator $F$ with trivial kernel and cokernel, using the limiting absorption principle an explicit set $\newcommand{\mbR}{{\mathbb R}}\newcommand{\LambHF}[1]{\Lambda(#1;F)}\LambHF{H_0} \subset \mbR$ of full Lebesgue measure is defined, such that for all $\newcommand{\LambHF}[1]{\Lambda(#1;F)}\lambda\in\LambHF{H_0+rV} \cap \LambHF{H_0},$ where $r \in\newcommand{\mbR}{{\mathbb R}} \mbR,$ the wave $w_\pm(\lambda; H_0+rV,H_0)$ and the scattering matrices $S(\lambda; H_0+rV,H_0)$ can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including the stationary formula for the scattering matrix. This version of abstract scattering theory allows us, in particular, to prove that $$\newcommand{\xia}{\xi^{(a)}}\newcommand{\mbR}{{\mathbb R}} \det S(\lambda;H_0+V,H_0) = e^{-2\pi i \xia(\lambda)}, \quad\ \text{a.e.} \ \lambda \in \mbR, $$ where $\newcommand{\xia}{\xi^{(a)}}\xia(\lambda) = \xia_{H_0+V,H_0}(\lambda)$ is the so called absolutely continuous part of the spectral shift function defined by $$ \newcommand{\xia}{\xi^{(a)}} \xia_{H_0+V,H_0}(\lambda) := \frac d{d\lambda} \int_0^1 \operatorname{Tr}(V E^{(a)}_{H_0+rV}(\lambda)) \,dr $$ and where $E_H^{(a)}(\lambda)=E^{(a)}_{(-\infty,\lambda)}(H)$ denotes the absolutely continuous part of the spectral projection. Combined with the Birman–Kreĭn formula, this implies that the singular part of the spectral shift function, $$\newcommand{\xis}{\xi^{(s)}} \xis_{H_0+V,H_0}(\lambda) := \frac d{d\lambda} \int_0^1 \operatorname{Tr}(V E^{(s)}_{H_0+rV}(\lambda)) \,dr, $$ is an almost everywhere integer-valued function, where $E_H^{(s)}(\lambda)=E^{(s)}_{(-\infty,\lambda)}(H)$ denotes the singular part of the spectral projection.

It is also shown that eigenvalues of the scattering matrix $S(\lambda;H_0+V,H_0)$ can be connected to $1$ in two natural ways, and that the singular spectral shift function measures the difference of the spectral flows of eigenvalues of the scattering matrix.