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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, Issue 2, Pages 3–28 (Mi vuu424)

MATHEMATICS

On the spectrum of a two-dimensional generalized periodic Schrödinger operator. II

L. I. Danilov

Physical Technical Institute, Ural Branch of the Russian Academy of Sciences, ul. Kirova, 132, Izhevsk, 426000, Russia

Abstract: The paper is concerned with the problem of absolute continuity of the spectrum of the two-dimensional generalized periodic Schrödinger operator $H_g+V=-\nabla g\nabla +V$ where the continuous positive function $g$ and the scalar potential $V$ have a common period lattice $\Lambda$. The solutions of the equation $(H_g+V)\varphi =0$ determine, in particular, the electric field and the magnetic field of electromagnetic waves propagating in two-dimensional photonic crystals. The function $g$ and the scalar potential $V$ are expressed in terms of the electric permittivity $\varepsilon$ and the magnetic permeability $\mu$ ($V$ also depends on the frequency of the electromagnetic wave). The electric permittivity $\varepsilon$ may be a discontinuous function (and usually it is chosen to be piecewise constant) so the problem to relax the known smoothness conditions on the function $g$ that provide absolute continuity of the spectrum of the operator $H_g+V$ arises. In the present paper we assume that the Fourier coefficients of the functions $g^{\pm \frac 12}$ for some $q\in [1,\frac 43 )$ satisfy the condition $\sum ( |N|^{\frac 12}|(g^{\pm \frac 12})_N|) ^q < +\infty$, and the scalar potential $V$ has relative bound zero with respect to the operator $-\Delta$ in the sense of quadratic forms. Let $K$ be the fundamental domain of the lattice $\Lambda$, and assume that $K^*$ is the fundamental domain of the reciprocal lattice $\Lambda ^*$. The operator $H_g+V$ is unitarily equivalent to the direct integral of operators $H_g(k)+V$, with quasimomenta $k\in 2\pi K^*$, acting on the space $L^2(K)$. The last operators can be also considered for complex vectors $k+ik^{\prime }\in {\mathbb C}^2$. We use the Thomas method. The proof of absolute continuity of the spectrum of the operator $H_g+V$ amounts to showing that the operators $H_g(k+ik^{\prime })+V-\lambda$, $\lambda \in {\mathbb R}$, are invertible for some appropriately chosen complex vectors $k+ik^{\prime }\in {\mathbb C}^2$ (depending on $g$, $V$, and the number $\lambda$) with sufficiently large imaginary parts $k^{\prime }$.

Keywords: generalized Schrödinger operator, absolute continuity of the spectrum, periodic potential.

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UDC: 517.958+517.984.5
MSC: 35P05

Citation: L. I. Danilov, “On the spectrum of a two-dimensional generalized periodic Schrödinger operator. II”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2014, no. 2, 3–28

Citation in format AMSBIB
\Bibitem{Dan14} \by L.~I.~Danilov \paper On the spectrum of a two-dimensional generalized periodic Schr\"{o}dinger operator. II \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2014 \issue 2 \pages 3--28 \mathnet{http://mi.mathnet.ru/vuu424} 

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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
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