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Singular Spectrum Near a Singular Point of Friedrichs Model Operators of Absolute Type

  • Serguei I. IakovlevEmail author
Article
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Abstract

In \(L_{2} {\left( \mathbb{R} \right)}\) we consider a family of selfadjoint operators of the Friedrichs model: \(A_{m} = {\left| t \right|}^{m} \cdot + V\). Here \({\left| t \right|}^{m} \cdot \) is the operator of multiplication by the corresponding function of the independent variable \(t \in \mathbb{R}\), and \(V\) (perturbation) is a trace-class integral operator with a continuous Hermitian kernel \(v{\left( {t,x} \right)}\) satisfying some smoothness condition. These absolute type operators have one singular point \(t = 0\) of order \(m > 0\). Conditions on the kernel \(v{\left( {t,x} \right)}\) are found guaranteeing the absence of the point spectrum and the singular continuous one of such operators near the origin. These conditions are actually necessary and sufficient. They depend on the finiteness of the rank of a perturbation operator and on the order of singularity \(m\). The sharpness of these conditions is confirmed by counterexamples.

Mathematics Subject Classifications (2000)

47B06 47B25 

Key words

analytic functions eigenvalues Friedrichs model linear system  modulus of continuity selfadjoint operators singular point zeros 

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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  1. 1.Departamento de MatematicasUniversidad Simon Bolivar CaracasVenezuela

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