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Ufa Mathematical Journal, 2020, Volume 12, Issue 4, Pages 64–77
DOI: https://doi.org/10.13108/2020-12-4-64
(Mi ufa536)
 

This article is cited in 3 scientific papers (total in 4 papers)

On localization conditions for spectrum of model operator for Orr–Sommerfeld equation

Kh. K. Ishkin, R. I. Marvanov

Bashkir State University, Zaki Validi str. 32, 450000, Ufa, Russia
References:
Abstract: For a model operator $L(\varepsilon)$ related with Orr-Sommerfeld equation, we study the necessity of known Shkalikov conditions sufficient for a localization of the spectrum at a graph of Y-shape. We consider two types of the potentials, for which an unbounded part $\Gamma_\infty$ of the limiting spectral graph (LSG) is constructed in an explicit form. The first of them is a piece-wise potential with countably many jumps. We show that if the discontinuity points of this potential converge rather fast to one of the end-points of the interval $(0,1)$, then $\Gamma_\infty$ consists in countably many rays. The second potential is glued from two holomorphic functions. We show that $\Gamma_\infty$ consists in two curves if the derivative at the gluing point has a jump and Langer conditions are satisfied in the domain enveloped by the Stokes lines ensuring the possibility of constructing WKB-expansions. If the gluing is infinitely differentiable, WKB-estimates are insufficient to clarify the spectral picture. Because of this we consider an inverse problem: given some spectral data, clarify analytic properties of the potential in the vicinity of the interval $(0,1)$. In order to understand the nature of spectral data, we first solve a direct problem extended to a complex $\varepsilon$-plane. It turns out that if we assume the holomorphy of the potential in the vicinity of the segment $[0,1]$, then for small $\varepsilon$ in the sector $\mathcal{E}$ of opening $\pi/2$, the part of the spectrum $L(\varepsilon)$ outside some circle satisfies quantizaion conditions of Bohr-Sommerfeld type. In the concluding part of the work we solve the inverse problem. As spectral data, quantization conditions obtained in the direct problem and taken in a slightly weaker form serve. We prove that if the potential is a monotone continuously differentiable function and the mentioned conditions are satisfied, then the potential admits an analytic continuation into some neighbourhood of the interval $(0,1)$. This proves the necessity of Shkalikov conditions at least in a local sense.
Keywords: Orr-Sommerfeld equation, localization of spectrum, limiting spectral graph.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-02-2020-1421/1
Russian Foundation for Basic Research 20-31-90999
The research of the first author is made in the framework of the development program of Scientific and Educational Mathematical Center of Privolzhsky Federal District, additional agreement no. 075-02-2020-1421/1 to agreement no. 075-02-2020-1421. The second author is supported by Russian Foundation for Basic Researches (project no. 20-31-90999).
Received: 20.10.2020
Russian version:
Ufimskii Matematicheskii Zhurnal, 2020, Volume 12, Issue 4, Pages 66–79
Bibliographic databases:
Document Type: Article
UDC: 517.984, 517.928
MSC: 47E05, 76E25
Language: English
Original paper language: Russian
Citation: Kh. K. Ishkin, R. I. Marvanov, “On localization conditions for spectrum of model operator for Orr–Sommerfeld equation”, Ufimsk. Mat. Zh., 12:4 (2020), 66–79; Ufa Math. J., 12:4 (2020), 64–77
Citation in format AMSBIB
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\by Kh.~K.~Ishkin, R.~I.~Marvanov
\paper On localization conditions for spectrum of model operator for Orr--Sommerfeld equation
\jour Ufimsk. Mat. Zh.
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\vol 12
\issue 4
\pages 66--79
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\transl
\jour Ufa Math. J.
\yr 2020
\vol 12
\issue 4
\pages 64--77
\crossref{https://doi.org/10.13108/2020-12-4-64}
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Linking options:
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  • https://doi.org/10.13108/2020-12-4-64
  • https://www.mathnet.ru/eng/ufa/v12/i4/p66
    Erratum
    • Erratum
      Kh. K. Ishkin, R. I. Marvanov
      Ufimsk. Mat. Zh., 2022, 14:4, 154
    This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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