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Singular Perturbations and Operators in Rigged Hilbert Spaces

Abstract

A notion of regularity and singularity for a special class of operators acting in a rigged Hilbert space \({\mathcal{D} \subset \mathcal{H}\subset \mathcal{D}^\times}\) is proposed and it is shown that each operator decomposes into a sum of a regular and a singular part. This property is strictly related to the corresponding notion for sesquilinear forms. A particular attention is devoted to those operators that are neither regular nor singular, pointing out that a part of them can be seen as perturbation of a self-adjoint operator on \({\mathcal{H}}\). Some properties for such operators are derived and some examples are discussed.

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Correspondence to Camillo Trapani.

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di Bella, S., Trapani, C. Singular Perturbations and Operators in Rigged Hilbert Spaces. Mediterr. J. Math. 13, 2011–2024 (2016). https://doi.org/10.1007/s00009-015-0590-7

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  • DOI: https://doi.org/10.1007/s00009-015-0590-7

Mathematics Subject Classification

  • 47L60
  • 47A70

Keywords

  • Regular operators
  • singular operators
  • rigged Hilbert spaces