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 Izv. IMI UdGU, 2018, Volume 51, Pages 3–41 (Mi iimi352)

On the spectrum of a two-dimensional schrödinger operator with a homogeneous magnetic field and a periodic electric potential

L. I. Danilov

Udmurt Federal Research Center of the Ural Branch of the Russian Academy of Sciences, ul. T. Baramzinoi, 34, Izhevsk, 426067, Russia

Abstract: We consider the two-dimensional Schrödinger operator $\widehat H_B+V$ with a uniform magnetic field $B$ and a periodic electric potential $V$. The absence of eigenvalues (of infinite multiplicity) in the spectrum of the operator $\widehat H_B+V$ is proved if the electric potential $V$ is a nonconstant trigonometric polynomial and the condition $(2\pi )^{-1} Bv(K)=Q^{-1}$ for the magnetic flux is fulfilled where $Q\in \mathbb{N}$ and the $v(K)$ is the area of the elementary cell $K$ of the period lattice $\Lambda \subset \mathbb{R}^2$ of the potential $V$. In this case the singular component of the spectrum is absent so the spectrum is absolutely continuous. In this paper, we use the magnetic Bloch theory. Instead of the lattice $\Lambda$ we choose the lattice $\Lambda _{ Q}=\{ N_1QE^1+N_2E^2:N_j\in \mathbb{Z} , j=1,2\}$ where $E^1$ and $E^2$ are basis vectors of the lattice $\Lambda$. The operator $\widehat H_B+V$ is unitarily equivalent to the direct integral of the operators $\widehat H_B(k)+V$ with $k\in 2\pi K_{ Q}^*$ acting on the space of magnetic Bloch functions where $K_{ Q}^*$ is an elementary cell of the reciprocal lattice $\Lambda _{ Q}^*\subset \mathbb{R}^2$. The proof of the absence of eigenvalues in the spectrum of the operator $\widehat H_B+V$ is based on the following assertion: if $\lambda$ is an eigenvalue of the operator $\widehat H_B+V$, then the $\lambda$ is an eigenvalue of the operators $\widehat H_B(k+i\varkappa )+V$ for all $k, \varkappa \in \mathbb{R}^2$ and, moreover, (under the assumed conditions on the $V$ and the $B$) there is a vector $k_0\in \mathbb{C}^2 \backslash \{0\}$ such that the eigenfunctions of the operators $\widehat H_B(k+\zeta k_0)+V$ with $\zeta \in \mathbb{C}$ are trigonometric polynomials $\sum \zeta ^j\Phi _j$ in $\zeta$.

Keywords: Schrödinger operator, spectrum, periodic electric potential, homogeneous magnetic field.

DOI: https://doi.org/10.20537/2226-3594-2018-51-01

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Bibliographic databases:

UDC: 517.958, 517.984.5
MSC: 35P05

Citation: L. I. Danilov, “On the spectrum of a two-dimensional schrödinger operator with a homogeneous magnetic field and a periodic electric potential”, Izv. IMI UdGU, 51 (2018), 3–41

Citation in format AMSBIB
\Bibitem{Dan18} \by L.~I.~Danilov \paper On the spectrum of a two-dimensional schrödinger operator with a homogeneous magnetic field and a periodic electric potential \jour Izv. IMI UdGU \yr 2018 \vol 51 \pages 3--41 \mathnet{http://mi.mathnet.ru/iimi352} \crossref{https://doi.org/10.20537/2226-3594-2018-51-01} \elib{https://elibrary.ru/item.asp?id=35269037} 

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This publication is cited in the following articles:
1. L. I. Danilov, “O spektre relyativistskogo gamiltoniana Landau s periodicheskim elektricheskim potentsialom”, Izv. IMI UdGU, 54 (2019), 3–26
2. L. I. Danilov, “Spectrum of the Landau Hamiltonian with a periodic electric potential”, Theoret. and Math. Phys., 202:1 (2020), 41–57
3. L. I. Danilov, “O spektre gamiltoniana Landau s periodicheskim elektricheskim potentsialom $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$”, Izv. IMI UdGU, 55 (2020), 42–59
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