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Russia Izhevsk
Year
2014
Issue
2
Pages
3-28
 Section Mathematics Title On the spectrum of a two-dimensional generalized periodic Schrödinger operator. II Author(-s) Danilov L.I.a Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa 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 $\sum \bigl( |N|^\frac 12|(g^\pm \frac 12)_N|\bigr) ^q < +\infty$, 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 two-dimensional generalized periodic Schrödinger operator. II, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 3-28. References Reed M., Simon B. Methods of modern mathematical physics, Vol. II: Fourier analysis. 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