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
$\lambda_j(p_1,p_2)$, $j=0,1,2,\dots$. 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.
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
\Bibitem{Lys85}
\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://mathscinet.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{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1985D277400005}
Linking options:
https://www.mathnet.ru/eng/tmf5144
https://www.mathnet.ru/eng/tmf/v65/i3/p368
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L. I. Danilov, “Spectrum of the Landau Hamiltonian with a periodic electric potential”, Theoret. and Math. Phys., 202:1 (2020), 41–57
Giuseppe De Nittis, Kyonori Gomi, Massimo Moscolari, “The geometry of (non-Abelian) Landau levels”, Journal of Geometry and Physics, 152 (2020), 103649
L. I. Danilov, “O spektre dvumernogo operatora Shredingera s odnorodnym magnitnym polem i periodicheskim elektricheskim potentsialom”, Izv. IMI UdGU, 51 (2018), 3–41
Gianluca Panati, Herbert Spohn, Stefan Teufel, Analysis, Modeling and Simulation of Multiscale Problems, 2006, 595
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
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
V.A. Geyler, “First Chern class of lattice magneto-Bloch bundles”, Reports on Mathematical Physics, 38:3 (1996), 333
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
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
Alexandrina Nenciu, G. Nenciu, “Existence of exponentially localized Wannier functions for nonperiodic systems”, Phys. Rev. B, 47:16 (1993), 10112
G. Nenciu, “Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians”, Rev. Mod. Phys., 63:1 (1991), 91
J. E. Avron, A. Raveh, B. Zur, “Adiabatic quantum transport in multiply connected systems”, Rev. Mod. Phys., 60:4 (1988), 873
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