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hal-00455207, version 1

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

Alexander Pushnitski, Dimitri Yafaev () 1

Journal of Functional Analysis 259, 8 (2010) 1950-1973

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 $(-\infty,\lambda)$. The purpose of the paper is to study the spectral properties of the difference $D(\lambda)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(\lambda)$ in terms of the eigenvalues of the scattering matrix $S(\lambda)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(\lambda)$ is empty.

• 1:  Institut de Recherche Mathématique de Rennes (IRMAR)
• CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne
• Domain : Mathematics/Spectral Theory
Mathematics/Functional Analysis
• Keywords : scattering matrix – Carleman operator – absolutely continuous spectrum – spectral projections
• Comment : 21 page – Latex

• hal-00455207, version 1
• oai:hal.archives-ouvertes.fr:hal-00455207
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• Submitted on: Tuesday, 9 February 2010 17:10:36
• Updated on: Tuesday, 12 June 2012 17:10:07