A. V. D'yachenko, A. A. Shkalikov, “On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile”, Funktsional. Anal. i Prilozhen., 36:3 (2002), 71–75; Funct. Anal. Appl., 36:3 (2002), 228

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Funktsional'nyi Analiz i ego Prilozheniya, 2002, Volume 36, Issue 3, Pages 71–75
DOI: https://doi.org/10.4213/faa208 (Mi faa208)

This article is cited in 15 scientific papers (total in 15 papers)

Brief communications

On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile

A. V. D'yachenko^a, A. A. Shkalikov^b

^a M. V. Lomonosov Moscow State University
^b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Full-text PDF (139 kB) Citations (15)

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DOI: https://doi.org/10.4213/faa208

Abstract: A model spectral problem of the form

$-i\varepsilon y''+xy=\lambda y$ on the finite interval

$[-1,1]$ with the Dirichlet boundary conditions is considered. Here

$\lambda$ is the spectral parameter and

$\varepsilon$ is positive. The behavior of the spectrum of this problem as

$\varepsilon\to 0$ is completely investigated. The limit curves are found to which the eigenvalues concentrate and the counting eigenvalue functions along these curves are obtained.

Keywords: the Airy function, Couette flow, quasiclassical eigenvalue formulas.

Received: 29.10.2001

English version:
Functional Analysis and Its Applications, 2002, Volume 36, Issue 3, Pages 228–232
DOI: https://doi.org/10.1023/A:1020158323673

Bibliographic databases:

Document Type: Article

UDC: 517.927+517.928

Language: Russian

Citation: A. V. D'yachenko, A. A. Shkalikov, “On a Model Problem for the Orr–Sommerfeld Equation with Linear Profile”, Funktsional. Anal. i Prilozhen., 36:3 (2002), 71–75; Funct. Anal. Appl., 36:3 (2002), 228–232

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/faa208

https://doi.org/10.4213/faa208

https://www.mathnet.ru/eng/faa/v36/i3/p71

This publication is cited in the following 15 articles:

D.I. Borisov, D.M. Polyakov, “Uniform Spectral Asymptotics for a Schrödinger Operator on a Segment with Delta-Interaction”, Russ. J. Math. Phys., 31:2 (2024), 149
D. I. Borisov, D. M. Polyakov, “Uniform asymptotics for eigenvalues of model Schrödinger operator with small translation”, Ufa Math. J., 16:3 (2024), 1–20
Shafarevich A., “Quantization Conditions on Riemannian Surfaces and Spectral Series of Non-Selfadjoint Operators”, Formal and Analytic Solutions of Diff. Equations, Springer Proceedings in Mathematics & Statistics, 256, eds. Filipuk G., Lastra A., Michalik S., Springer, 2018, 177–187
A. M. Savchuk, A. A. Shkalikov, “Spectral Properties of the Complex Airy Operator on the Half-Line”, Funct. Anal. Appl., 51:1 (2017), 66–79
D. V. Nekhaev, A. I. Shafarevich, “A quasiclassical limit of the spectrum of a Schrödinger operator with complex periodic potential”, Sb. Math., 208:10 (2017), 1535–1556
Kh. K. Ishkin, “Localization criterion for the spectrum of the Sturm–Liouville operator on a curve”, St. Petersburg Math. J., 28:1 (2017), 37–63
A. I. Esina, A. I. Shafarevich, “Asymptotics of the Spectrum and Eigenfunctions of the Magnetic Induction Operator on a Compact Two-Dimensional Surface of Revolution”, Math. Notes, 95:3 (2014), 374–387
Esina A.I., Shafarevich A.I., “Analogs of Bohr-Sommerfeld-Maslov Quantization Conditions on Riemann Surfaces and Spectral Series of Nonself-Adjoint Operators”, Russ. J. Math. Phys., 20:2 (2013), 172–181
Ortiz de Zarate J.M., Sengers J.V., “Hydrodynamic Fluctuations in Laminar Fluid Flow. I. Fluctuating Orr-Sommerfeld Equation”, J Stat Phys, 144:4 (2011), 774–792
A. I. Esina, A. I. Shafarevich, “Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schrödinger Operator with Complex Potential”, Math. Notes, 88:2 (2010), 209–227
Malevich, AE, “Couette flow in channels with wavy walls”, Acta Mechanica, 197:3–4 (2008), 247
de Zarate, JMO, “Transverse-velocity fluctuations in a liquid under steady shear”, Physical Review E, 77:2 (2008), 026306
Gunther, U, “MHD alpha(2)-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator”, Journal of Mathematical Physics, 46:6 (2005), 063504
A. A. Shkalikov, “Spectral Portraits of the Orr–Sommerfeld Operator with Large Reynolds Numbers”, Journal of Mathematical Sciences, 124:6 (2004), 5417–5441
S. N. Tumanov, A. A. Shkalikov, “On the Spectrum Localization of the Orr–Sommerfeld Problem for Large Reynolds Numbers”, Math. Notes, 72:4 (2002), 519–526

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Functional Analysis and Its Applications

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