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 TMF, 2002, Volume 131, Number 2, Pages 304–331 (Mi tmf332)

The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field

J. Brüninga, S. Yu. Dobrokhotovb, K. V. Pankrashinba

a Humboldt University
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: The asymptotic form of the bottom part of the spectrum of the two-dimensional magnetic Schrödinger operator with a periodic potential in a strong magnetic field is studied in the semiclassical approximation. Averaging methods permit reducing the corresponding classical problem to a one-dimensional problem on the torus; we thus show the almost integrability of the original problem. Using elementary corollaries from the topological theory of Hamiltonian systems, we classify the almost invariant manifolds of the classical Hamiltonian. The manifolds corresponding to the bottom part of the spectrum are closed or nonclosed curves and points. Their geometric and topological characteristics determine the asymptotic form of parts of the spectrum (spectral series). We construct this asymptotic form using the methods of the semiclassical approximation with complex phases. We discuss the relation of the asymptotic form obtained to the magneto-Bloch conditions and asymptotics of the band spectrum.

DOI: https://doi.org/10.4213/tmf332

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English version:
Theoretical and Mathematical Physics, 2002, 131:2, 704–728

Bibliographic databases:

Citation: J. Brüning, S. Yu. Dobrokhotov, K. V. Pankrashin, “The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field”, TMF, 131:2 (2002), 304–331; Theoret. and Math. Phys., 131:2 (2002), 704–728

Citation in format AMSBIB
\Bibitem{BruDobPan02} \by J.~Br\"uning, S.~Yu.~Dobrokhotov, K.~V.~Pankrashin \paper The Asymptotic Form of the Lower Landau Bands in a Strong Magnetic Field \jour TMF \yr 2002 \vol 131 \issue 2 \pages 304--331 \mathnet{http://mi.mathnet.ru/tmf332} \crossref{https://doi.org/10.4213/tmf332} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1932256} \zmath{https://zbmath.org/?q=an:1039.81023} \transl \jour Theoret. and Math. Phys. \yr 2002 \vol 131 \issue 2 \pages 704--728 \crossref{https://doi.org/10.1023/A:1015433000783} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000176246100011} 

• http://mi.mathnet.ru/eng/tmf332
• https://doi.org/10.4213/tmf332
• http://mi.mathnet.ru/eng/tmf/v131/i2/p304

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

This publication is cited in the following articles:
1. Bruning, J, “The spectral asymptotics of the two-dimensional Schrodinger operator with a strong magnetic field. II”, Russian Journal of Mathematical Physics, 9:4 (2002), 400
2. J. Brüning, S. Yu. Dobrokhotov, V. A. Geiler, K. Pankrashkin, “Hall conductivity of minibands lying at the wings of Landau levels”, JETP Letters, 77:11 (2003), 616–618
3. Pankrashkin K, “On semiclassical dispersion relations of Harper-like operators”, Journal of Physics A-Mathematical and General, 37:48 (2004), 11681–11698
4. A. Yu. Anikin, J. Brüning, S. Yu. Dobrokhotov, “Averaging and trajectories of a Hamiltonian system appearing in graphene placed in a strong magnetic field and a periodic potential”, J. Math. Sci., 223:6 (2017), 656–666
5. Yu. A. Kordyukov, I. A. Taimanov, “Kvaziklassicheskoe priblizhenie dlya magnitnykh monopolei”, UMN, 75:6(456) (2020), 85–106
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