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TMF, 1985, Volume 65, Number 3, Pages 368–378 (Mi tmf5144)  

This article is cited in 5 scientific papers (total in 5 papers)

Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential

A. S. Lyskova


Abstract: A study is made of the two-dimensional Schrödinger operator $H$ in a periodic magnetic field $B(x,y)$ and in an electric field with periodic potential $V(x,y)$. It is assumed that the functions $B(x,y)$ and $V(x,y)$ are periodic with respect to some lattice $\Gamma$ in $R^2$ and that the magnetic flux through a unit cell is an integral number. The operator $H$ is represented as a direct integral over the two-dimensional torus of the reciprocal lattice of elliptic self-adjoint operators $H_{p_1,p_2}$, which possess a discrete spectrum $\lambda_j(p_1,p_2)$, $j=0,1,2,…$. On the basis of an exactly integrable case – the Schrödinger operator in a constant magnetic field – perturbation theory is used to investigate the typical dispersion laws $\lambda_j(p_1,p_2)$ and establish their topological characteristics (quantum numbers). A theorem is proved: In the general case, the Schrödinger operator has a countable number of dispersion laws with arbitrary quantum numbers in no way related to one another or to the flux of the external magnetic field.

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English version:
Theoretical and Mathematical Physics, 1985, 65:3, 1218–1225

Bibliographic databases:

Received: 03.12.1984

Citation: A. S. Lyskova, “Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential”, TMF, 65:3 (1985), 368–378; Theoret. and Math. Phys., 65:3 (1985), 1218–1225

Citation in format AMSBIB
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\by A.~S.~Lyskova
\paper Topological characteristics of the spectrum of the Schr\"odinger operator in a~magnetic field and in a~weak potential
\jour TMF
\yr 1985
\vol 65
\issue 3
\pages 368--378
\mathnet{http://mi.mathnet.ru/tmf5144}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=829903}
\transl
\jour Theoret. and Math. Phys.
\yr 1985
\vol 65
\issue 3
\pages 1218--1225
\crossref{https://doi.org/10.1007/BF01036130}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1985D277400005}


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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. O. M. Ivanov, A. G. Savinkov, “Nontrivial $U(1)$ bundles over tori and properties of many-particle systems with topological charge”, Theoret. and Math. Phys., 96:1 (1993), 806–817  mathnet  crossref  mathscinet  zmath  isi
    2. V. A. Geiler, V. V. Demidov, “Spectrum of three-dimensional landau operator perturbed by a periodic point potential”, Theoret. and Math. Phys., 103:2 (1995), 561–569  mathnet  crossref  mathscinet  zmath  isi
    3. 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  isi
    4. Yu. P. Chuburin, “The Spectrum and Eigenfunctions of the Two-Dimensional Schrödinger Operator with a Magnetic Field”, Theoret. and Math. Phys., 134:2 (2003), 212–221  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. L. I. Danilov, “O spektre dvumernogo operatora Shredingera s odnorodnym magnitnym polem i periodicheskim elektricheskim potentsialom”, Izv. IMI UdGU, 51 (2018), 3–41  mathnet  crossref  elib
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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