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Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces

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Abstract

Let A be an unbounded self-adjoint operator in a Hilbert separable space \(H_0\) with rigging \(H_ - \sqsupset H_0 \sqsupset H_ +\) such that \(D(A) = H_ +\) in the graph norm (here, \(D(A)\) is the domain of definition of A). Assume that \(H_ +\) is decomposed into the orthogonal sum \(H_ + = M \oplus N_ +\) so that the subspace \(M_ +\) is dense in \(H_0\). We construct and study a singularly perturbed operator A associated with a new rigging \(H_ - \sqsupset H_0 \sqsupset \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ +\), where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ + = M_ + = D(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} )\), and establish the relationship between the operators A and A.

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv, Ukraine

    R. V. Bozhok & V. D. Koshmanenko

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  1. R. V. Bozhok
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  2. V. D. Koshmanenko
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 622–632, May, 2005.

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Bozhok, R.V., Koshmanenko, V.D. Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces. Ukr Math J 57, 738–750 (2005). https://doi.org/10.1007/s11253-005-0224-5

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  • DOI: https://doi.org/10.1007/s11253-005-0224-5

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