A. I. Esina, A. I. Shafarevich, “Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schrödinger Operator with Complex Potential”, Mat. Zametki, 88:2 (2010), 229–248; Math. Notes, 88:2 (2010), 209

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Matematicheskie Zametki, 2010, Volume 88, Issue 2, Pages 229–248
DOI: https://doi.org/10.4213/mzm8803 (Mi mzm8803)

This article is cited in 16 scientific papers (total in 16 papers)

Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schrödinger Operator with Complex Potential

A. I. Esina^a, A. I. Shafarevich^bc

^a Moscow Institute of Physics and Technology
^b M. V. Lomonosov Moscow State University
^c A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Full-text PDF (644 kB) Citations (16)

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DOI: https://doi.org/10.4213/mzm8803

Abstract: We describe the asymptotics of the spectrum of the operator
$$ \widehat H\biggl(x,-\imath h\frac{\partial}{\partial x}\biggr)=-h^2\frac{\partial^2}{\partial x^2}+\imath(\cos x+\cos2x) $$
as $h\to0$ and show that the spectrum concentrates near some graph on the complex plane. We obtain equations for the eigenvalues, which are conditions on the periods of a holomorphic form on the corresponding Riemannian surface.

Keywords: Schrödinger operator, semiclassical spectrum of an operator, Riemannian surface, quantization condition, holomorphic form, Stokes line, monodromy matrix, turning point.

Received: 25.11.2009

English version:
Mathematical Notes, 2010, Volume 88, Issue 2, Pages 209–227
DOI: https://doi.org/10.1134/S0001434610070205

Bibliographic databases:

Document Type: Article

UDC: 517.984.55+514.84

Language: Russian

Citation: A. I. Esina, A. I. Shafarevich, “Quantization Conditions on Riemannian Surfaces and the Semiclassical Spectrum of the Schrödinger Operator with Complex Potential”, Mat. Zametki, 88:2 (2010), 229–248; Math. Notes, 88:2 (2010), 209–227

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mzm8803

https://doi.org/10.4213/mzm8803

https://www.mathnet.ru/eng/mzm/v88/i2/p229

This publication is cited in the following 16 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
A.A. Arzhanov, S.A. Stepin, V.A. Titov, V.V. Fufaev, “Stokes Phenomenon and Spectral Locus in a Problem of Singular Perturbation Theory”, Russ. J. Math. Phys., 31:3 (2024), 351
Stepin S.A., Fufaev V.V., “Wkb Asymptotics and Spectral Deformation in Semi-Classical Limit”, J. Dyn. Control Syst., 26:1 (2020), 175–198
Stepin S.A., Fufaev V.V., “Spectral Deformation in a Problem of Singular Perturbation Theory”, Dokl. Math., 99:1 (2019), 60–63
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
A. A. Shkalikov, S. N. Tumanov, “Spectral Portraits in the Semi-Classical Approximation of the Sturm-Liouville Problem with a Complex Potential”, J. Phys.: Conf. Ser., 1141 (2018), 012155
S. A. Stepin, V. V. Fufaev, “The phase-integral method in a problem of singular perturbation theory”, Izv. Math., 81:2 (2017), 359–390
V. V. Fufaev, “Level lines of harmonic functions related to some Abelian integrals”, Moscow University Mathematics Bulletin, 72:1 (2017), 15–23
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
A. Ifa, N. M'Hadbi, M. Rouleux, “On Generalized Bohr–Sommerfeld Quantization Rules for Operators with PT Symmetry”, Math. Notes, 99:5 (2016), 676–684
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
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
Tobias Gulden, Michael Janas, Alex Kamenev, “Riemann surface dynamics of periodic non-Hermitian Hamiltonians”, J. Phys. A: Math. Theor., 47:8 (2014), 085001
V. P. Maslov, “Bose–Einstein-Type Distribution for Nonideal Gas. Two-Liquid Model of Supercritical States and Its Applications”, Math. Notes, 94:2 (2013), 231–237
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
Maslov V.P., Maslova T.V., “Parastatistics and Phase Transition From a Cluster as a Fluctuation to a Cluster as a Distinguishable Object”, Russ. J. Math. Phys., 20:4 (2013), 468–475

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

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