Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 65 (1985) no. 3, pp. 368-378
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.
@article{TMF_1985_65_3_a4,
author = {A. S. Lyskova},
title = {Topological characteristics of the spectrum of the {Schr\"odinger} operator in a~magnetic field and in a~weak potential},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {368--378},
year = {1985},
volume = {65},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_1985_65_3_a4/}
}
TY - JOUR AU - A. S. Lyskova TI - Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential JO - Teoretičeskaâ i matematičeskaâ fizika PY - 1985 SP - 368 EP - 378 VL - 65 IS - 3 UR - http://geodesic.mathdoc.fr/item/TMF_1985_65_3_a4/ LA - ru ID - TMF_1985_65_3_a4 ER -
%0 Journal Article %A A. S. Lyskova %T Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential %J Teoretičeskaâ i matematičeskaâ fizika %D 1985 %P 368-378 %V 65 %N 3 %U http://geodesic.mathdoc.fr/item/TMF_1985_65_3_a4/ %G ru %F TMF_1985_65_3_a4
A. S. Lyskova. Topological characteristics of the spectrum of the Schrödinger operator in a magnetic field and in a weak potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 65 (1985) no. 3, pp. 368-378. http://geodesic.mathdoc.fr/item/TMF_1985_65_3_a4/
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