Singularly perturbed rank one linear operators

  • M.E. Dudkin National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute
  • O. Yu. Dyuzhenkova National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute Kyiv, Ukraine
Keywords: singularly perturbed operator; scale of Hilbert spaces; non-symmetric perturbations; eigenvalue; eigenvector; Krein’s formula

Abstract

The basic principles of the theory of singularly perturbed self-adjoint operators
are generalized to the case of closed linear operators with non-symmetric perturbation of rank one.
Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.
The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.
Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.
The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.
In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.
Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.
Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.

Author Biographies

M.E. Dudkin, National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute

Physics and Mathematics Faculty

Head of the Department of Differential Equations

 

O. Yu. Dyuzhenkova, National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute Kyiv, Ukraine

National Technical University of Ukraine
Igor Sikorsky Kyiv Polytechnic Institute
Kyiv, Ukraine

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Published
2021-12-26
How to Cite
Dudkin, M., & Dyuzhenkova, O. Y. (2021). Singularly perturbed rank one linear operators. Matematychni Studii, 56(2), 162-175. https://doi.org/10.30970/ms.56.2.162-175
Section
Articles