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Operator Theory in Inner Product Spaces pp 169–191Cite as

Spectral Analysis of Differential Operators with Indefinite Weights and a Local Point Interaction

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 175))

Abstract

We consider quasi-self-adjoint extensions of the symmetric operator \( A = - (\operatorname{sgn} x)\frac{{d^2 }} {{dx^2 }},dom(A) = \{ f \in W_2^2 (\mathbb{R}):f(0) = f'(0) = 0\} \), in the Hilbert space L 2(ℝ). The main result is a criterion of similarity to a normal operator for operators of this class. The spectra and resolvents of these extensions are described. As an application we describe the main spectral properties of the operators \( (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta } \right)and (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta '} \right) \).

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Author information

Authors and Affiliations

  1. Department of Mathematics, Donetsk National University, Universitetskaja 24, 83055, Donetsk, Ukraine

    Ilia Karabash & Aleksey Kostenko

Authors
  1. Ilia Karabash
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  2. Aleksey Kostenko
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Editor information

Editors and Affiliations

  1. Institut für Mathematik, MA 6-4, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin

    Karl-Heinz Förster , Peter Jonas  & Carsten Trunk ,  & 

  2. Mathematik, Technische Universität Wien, Wiedner Hauptstrasse 8-10/1411, A-1040, Wien

    Heinz Langer

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Karabash, I., Kostenko, A. (2007). Spectral Analysis of Differential Operators with Indefinite Weights and a Local Point Interaction. In: Förster, KH., Jonas, P., Langer, H., Trunk, C. (eds) Operator Theory in Inner Product Spaces. Operator Theory: Advances and Applications, vol 175. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8270-4_10

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