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

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 $k^{\prime }$.