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Russia Izhevsk
Section  Mathematics 
Title  On the spectrum of a twodimensional generalized periodic Schrödinger operator. II 
Author(s)  Danilov L.I.^{a} 
Affiliations  Physical Technical Institute, Ural Branch of the Russian Academy of Sciences^{a} 
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 \bigl( N^\frac 12(g^\pm \frac 12)_N\bigr) ^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 
UDC  517.958, 517.984.5 
MSC  35P05 
DOI  10.20537/vm140201 
Received  28 February 2014 
Language  Russian 
Citation  Danilov L.I. On the spectrum of a twodimensional generalized periodic Schrödinger operator. II, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 328. 
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