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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:

Document Type: Article
UDC: 517.958, 517.984.5
MSC: 35P05
Received: 18.04.2018

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{http://elibrary.ru/item.asp?id=35269037}


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