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arXiv · 0907.1518

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

Alexander Pushnitski·Dmitri Yafaev
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Abstract

In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,λ)$. The purpose of the paper is to study the spectral properties of the difference $D(λ)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(λ)$ in terms of the eigenvalues of the scattering matrix $S(λ)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(λ)$ is empty.

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Alexander Pushnitski, Dmitri Yafaev. 2009-07-09. Spectral theory of discontinuous functions of self-adjoint operators and scattering theory. https://arxiv.org/abs/0907.1518

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