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Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 4, Pages 177–204 (Mi izv399)  

This article is cited in 17 scientific papers (total in 17 papers)

On the limit behaviour of the spectrum of a model problem for the Orr–Sommerfeld equation with Poiseuille profile

S. N. Tumanov, A. A. Shkalikov

Abstract: This paper deals with a problem on the limiting behaviour of the spectra of the operators $L(\varepsilon)=i\varepsilon y^{\prime\prime}+x^2y$ with Dirichlet boundary conditions on a finite interval as the positive parameter $\varepsilon$ tends to zero. It is proved that the spectrum is concentrated along three curves in the complex plane. These curves connect a knot-point $\lambda_0$, which lies in the numerical range of the operator, with the points 0, 1 and $-i\infty$. We find uniform (with respect to $\varepsilon$) quasiclassical formulae for the distribution of the eigenvalues along these curves.

DOI: https://doi.org/10.4213/im399

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English version:
Izvestiya: Mathematics, 2002, 66:4, 829–856

Bibliographic databases:

UDC: 517.927+517.928
MSC: 34L20, 34B24, 76E15
Received: 04.07.2001

Citation: S. N. Tumanov, A. A. Shkalikov, “On the limit behaviour of the spectrum of a model problem for the Orr–Sommerfeld equation with Poiseuille profile”, Izv. RAN. Ser. Mat., 66:4 (2002), 177–204; Izv. Math., 66:4 (2002), 829–856

Citation in format AMSBIB
\by S.~N.~Tumanov, A.~A.~Shkalikov
\paper On the limit behaviour of the spectrum of a model problem for the Orr--Sommerfeld equation with Poiseuille profile
\jour Izv. RAN. Ser. Mat.
\yr 2002
\vol 66
\issue 4
\pages 177--204
\jour Izv. Math.
\yr 2002
\vol 66
\issue 4
\pages 829--856

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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. A. Shkalikov, “Spectral Portraits of the Orr–Sommerfeld Operator with Large Reynolds Numbers”, Journal of Mathematical Sciences, 124:6 (2004), 5417–5441  mathnet  crossref  mathscinet  zmath
    3. S. V. Galtsev, A. I. Shafarevich, “Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients”, Theoret. and Math. Phys., 148:2 (2006), 1049–1066  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. S. V. Galtsev, A. I. Shafarevich, “Spectrum and Pseudospectrum of non-self-adjoint Schrödinger Operators with Periodic Coefficients”, Math. Notes, 80:3 (2006), 345–354  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. V. I. Pokotilo, “Semiclassical Approximation for the Non-Self-Adjoint Sturm–Liouville Problem with the Potential $q(x)=x^4-a^2x^2$”, Math. Notes, 85:5 (2009), 755–759  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. V. I. Pokotilo, A. A. Shkalikov, “Semiclassical Approximation for a Nonself-Adjoint Sturm–Liouville Problem with a Parabolic Potential”, Math. Notes, 86:3 (2009), 442–446  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    8. Georgievskii D.V., Mueller W.H., Abali B.E., “Eigenvalue problems for the generalized Orr-Sommerfeld equation in the theory of hydrodynamic stability”, Doklady Physics, 56:9 (2011), 494–497  crossref  mathscinet  adsnasa  isi  elib  scopus
    9. Georgievskii D.V., Myuller V.Kh., Abali B.E., “Zadachi na sobstvennye znacheniya dlya obobschennogo uravneniya orra–zommerfelda v teorii gidrodinamicheskoi ustoichivosti”, Doklady akademii nauk, 440:1 (2011), 52–55  mathscinet  elib
    10. 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  crossref  mathscinet  zmath  isi  elib  scopus
    11. 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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. Tumanov S.N. Shkalikov A.A., “the Limit Spectral Graph in Semiclassical Approximation For the Sturm-Liouville Problem With Complex Polynomial Potential”, Dokl. Math., 92:3 (2015), 773–777  crossref  mathscinet  zmath  isi  scopus
    13. Vukadinovic J., Dedits E., Poje A.C., Schaefer T., “Averaging and Spectral Properties For the 2D Advection-Diffusion Equation in the Semi-Classical Limit For Vanishing Diffusivity”, Physica D, 310 (2015), 1–18  crossref  mathscinet  zmath  adsnasa  isi  scopus
    14. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    15. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    16. 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, ed. Filipuk G. Lastra A. Michalik S., Springer, 2018, 177–187  crossref  isi
    17. Tumanov S., “Completeness Theorem For the System of Eigenfunctions of the Complex Schrodinger Operator l-C = -D(2)/Dx(2) + Cx(2/3)”, J. Funct. Anal., 280:7 (2021), 108820  crossref  mathscinet  isi
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