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Analogs of Bohr-Sommerfeld-Maslov quantization conditions on Riemann surfaces and spectral series of nonself-adjoint operators

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Abstract

In the paper, the semiclassical asymptotic behavior of the eigenvalues of some nonself-adjoint operators important for applications is studied (for the Sturm-Liouville operator with complex potential and the operator of induction). It turns out that the asymptotic behavior can be calculated using the quantization conditions, which can be represented as the condition that the integrals of a holomorphic form over the cycles on the corresponding complex Lagrangian manifold, which is a Riemann surface of constant energy, are integers. In contrast to the real case (the Bohr-Sommerfeld-Maslov formulas), to calculate a chosen spectral series, it is sufficient to assume that the integral over only one of the cycles takes integer values, and different cycles determine different series.

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References

  1. A. A. Arzhanov and S. A. Stepin, “Semiclassical Spectral Asymptotics and the Stokes Phenomenon for the Weber Equation,” Dokl. Akad. Nauk 378(1), 18–21 (2001) [Dokl. Math. 63 (3), 306–309 (2001)].

    MathSciNet  Google Scholar 

  2. S. V. Galtsev and A. I. Shafarevich, “Spectrum and Pseudospectrum of Nonself-adjoint Schrödinger Operators with Periodic Coefficients,” Mat. Zametki 80(3), 456–466 (2006) [Math. Notes 80 (3), 356–366 (2006)].

    Article  MathSciNet  Google Scholar 

  3. S. V. Galtsev and A. I. Shafarevich, “Quantized Riemann Surfaces and Semiclassical Spectral Series for a Nonself-adjoint Schrödinger Operator with Periodic Coefficients,” Teor. Mat. Fiz. 148(2), 206–226 (2006) [Theor. Math. Phys. 148 (2), 1049–1066 (2006)].

    Article  MathSciNet  Google Scholar 

  4. I. Ts.[C.] Gohberg [Gokhberg] and M. G. Krein, Introduction to the Theory of Linear Nonself-adjoint Operators (Nauka, Moscow, 1965, American Mathematical Society, Providence, R.I., 1969).

    Google Scholar 

  5. A. V. D’yachenko and A. A. Shkalikov, “On a Model Problem for the Orr-Sommerfeld Equation with Linear Profile,” Funktsional. Anal. i Prilozhen. 36(3), 71–75 (2002) [Funct. Anal. Appl. 36 (3), 228–232 (2002)].

    Article  MathSciNet  Google Scholar 

  6. A. I. Esina and A. I. Shafarevich, “Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schrödinger Operator with Complex Potential,” Mat. Zametki 88(2), 61–79 (2010) [Math. Notes 88 (2), 209–227 (2010)].

    MathSciNet  Google Scholar 

  7. M. A. Evgrafov and M. V. Fedorjuk [Fedoryuk], “Asymptotic Behavior of Solutions of the Equation w″ − p(z,λ)w = 0 as λ → ∞ in the Complex z-Plane,” Uspekhi Mat. Nauk 21(1), 3–50 (1966) [in Russian].

    MathSciNet  MATH  Google Scholar 

  8. Ya. B. Zel’dovich and A. A. Ruzmaikin, “The Hydromagnetic Dynamo as the Source of Planetary, Solar, and Galactic Magnetism,” Uspekhi Fiz. Nauk 152(2), 263–284 (1987) [Sov. Phys. Usp. 30 (6), 494–506 (1987)].

    Article  ADS  Google Scholar 

  9. V. P. Maslov, Asymptotic Methods and Perturbation Theory (Izd-vo MGU, 1965).

    Google Scholar 

  10. S.-A. Stepin, “Nonself-adjoint Singular Perturbations: a Model of the Passage from a Discrete Spectrum to a Continuous Spectrum,” Uspekhi Mat. Nauk 50(6), 219–220 (1995) [Russ. Math. Surv. 50 (6), 1311–1313 (1995)].

    MathSciNet  Google Scholar 

  11. A. A. Shkalikov and S. N. Tumanov, “On the Limit Behaviour of the Spectrum of a Model Problem for the Orr-Sommerfeld Equation with Poiseuille Profile,” Izv. Ross. Akad. Nauk Ser. Mat. 66(4), 174–204 (2002) [Izv. Math. 66 (4), 829–856 (2002)].

    MathSciNet  Google Scholar 

  12. M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations (Izd. “Nauka,” Moscow, 1983; Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993).

    MATH  Google Scholar 

  13. A. A. Shkalikov, “On the Limit Behavior of the Spectrum for Large Values of the Parameter of a Model Problem,” Mat. Zametki 62(6), 950–953 (1997) [Math. Notes 62 (5–6), 796–799 (1997) (1998)].

    Article  MathSciNet  Google Scholar 

  14. E. B. Davies, “Pseudospectra of Differential Operators,” Operator Theory 43, 243–262 (2000).

    MathSciNet  MATH  Google Scholar 

  15. L. N. Trefethen, “Pseudospectra of Linear Operators,” ISIAM 95: Proceedings of the Third Int. Congress of Industrial and Applied Math., Academic Verlag, Berlin 401–434 (1996).

    Google Scholar 

  16. R. G. Drazin and W. H. Reid, Hydrodynamic Stability (Cambridge, 1981).

    MATH  Google Scholar 

  17. H. Roohian and A. I. Shafarevich, “Semiclassical Asymptotics of the Spectrum of a Nonself-adjoint Operator on the Sphere,” Russ. J. Math. Phys. 16(2), 309–315 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Roohian and A. I. Shafarevich, “Semiclassical Asymptotic Behavior of the Spectrum of a Nonselfadjoint Elliptic Operator on a Two-Dimensional Surface of Revolution,” Russ. J. Math. Phys. 17(3), 328–334 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. S. A. Stepin and V. A. Titov, “On the Concentration of Spectrum in the Model Problem of Singular Perturbation Theory,” Dokl. Akad. Nauk 413(1), 27–30 (2007) ([Dokl. Math. 75 (2), 197–200 (2007)].

    MathSciNet  Google Scholar 

  20. A. A. Shkalikov, “Spectral Portraits of the Orr-Sommerfeld Operator with Large Reynolds Numbers,” Sovrem. Mat. Fundam. Napravl. 3, 89–112, electronic only (2003) [J. Math. Sci., New York 124 (6), 5417–5441 (2004)].

    Google Scholar 

  21. V. I. Pokotilo and A. A. Shkalikov, “Semiclassical Approximation for a Nonself-adjoint Sturm-Liouville Problem with a Parabolic Potential,” Mat. Zametki 86(3), 469–473 (2009) [Math. Notes 86 (3), 442–446 (2009)].

    Article  MathSciNet  Google Scholar 

  22. L. K. Kusainova, A. Zh. Monashova, and A. A. Shkalikov, “Asymptotics of the Eigenvalues of the Second-Order Nonself-adjoint Differential Operator on the Axis,” Mat. Zametki 93(4), 630–633 (2013) [to appear in Math. Notes].

    Article  Google Scholar 

  23. V. I. Arnold, Ya. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokolov, “A Magnetic Field in a Stationary Flow with Stretching in Riemannian Space,” Zh. Eksp. Teor. Fiz. 81, 2052–2058 (1981).

    Google Scholar 

  24. A. V. Bolsinov and I. A. Taimanov, “Integrable Geodesic Flows with Positive Topological Entropy,” Invent. Math. 140, 639–650 (2000).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. A. V. Bolsinov, H. R. Dullin, and A. P. Veselov, “Spectra of Sol-Manifolds: Arithmetic and Quantum Monodromy,” Comm. Math. Phys. 264, 583–611 (2006).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. A. I. Esina and A. I. Shafarevich, “Asymptotics of the Spectrum and the Eigenfunctions of the Operator of Magnetic Induction on a Two-Dimensional Compact Surface of Revolution” (in print).

  27. V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximation for the Equations of Quantum Mechanics (Izdat. “Nauka,” Moscow, 1976) [in Russian].

    MATH  Google Scholar 

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Authors
  1. A. I. Esina
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  2. A. I. Shafarevich
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Additional information

The research was supported by the Grant of the Government of the Russian Federation for the State support of scientific researches carried out under the supervision of leading scientists in the Federal State Budget Educational Institution of Higher Professional Education “Lomonosov Moscow State University” according to the agreement no. 11.G34.31.0054, and also by the Russian Foundation for Basic Research under grants nos. 09-01-12063-ofi-m, 11-01-00937-a, and 13-01-00664, by the Program of supporting leasing scientific schools (under grant no.-3224.2010.1), and a grant for supporting young scientists “My First Grant” no. 12-01-31235.

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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, 172–181 (2013). https://doi.org/10.1134/S1061920813020052

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  • DOI: https://doi.org/10.1134/S1061920813020052

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