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




This article is cited in 1 scientific paper (total in 1 paper)
MATHEMATICS
On the spectrum of a twodimensional 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 twodimensional 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 twodimensional 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 Received: 28.02.2014
Citation:
L. I. Danilov, “On the spectrum of a twodimensional 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 twodimensional generalized periodic Schr\"{o}dinger operator. II
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2014
\issue 2
\pages 328
\mathnet{http://mi.mathnet.ru/vuu424}
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This publication is cited in the following articles:

L. I. Danilov, “O spektre relyativistskogo gamiltoniana Landau s periodicheskim elektricheskim potentsialom”, Izv. IMI UdGU, 54 (2019), 3–26

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