Spectrum of the~Landau Hamiltonian with a~periodic electric potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 1, pp. 47-65
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We define a class of periodic electric potentials for which the spectrum of
the two-dimensional Schrödinger operator is absolutely continuous in the case of a homogeneous magnetic field $B$ with a rational flux $\eta=
(2\pi)^{-1}Bv(K)$, where $v(K)$ is the area of an elementary cell $K$ in the lattice of potential periods. Using properties of functions in this class,
we prove that in the space of periodic electric potentials in $L^2_{\mathrm{loc}}(\mathbb R^2)$ with a given period lattice and identified with $L^2(K)$, there
exists a second-category set (in the sense of Baire) such that for any
electric potential in this set and any homogeneous magnetic field with a rational flow $\eta$, the spectrum of the two-dimensional Schrödinger
operator is absolutely continuous.
Keywords:
two-dimensional Schrödinger operator, absolute spectrum continuity, periodic potential, homogeneous magnetic field.
@article{TMF_2020_202_1_a4,
author = {L. I. Danilov},
title = {Spectrum of {the~Landau} {Hamiltonian} with a~periodic electric potential},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {47--65},
publisher = {mathdoc},
volume = {202},
number = {1},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2020_202_1_a4/}
}
L. I. Danilov. Spectrum of the~Landau Hamiltonian with a~periodic electric potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 1, pp. 47-65. http://geodesic.mathdoc.fr/item/TMF_2020_202_1_a4/