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Resonance and double negative behavior in metamaterials. (English) Zbl 1301.35165
A transmission eigenvalue problem in a unit square with periodic boundary conditions is considered for an operator of the form \(-\nabla\cdot\gamma\nabla\), where the scalar multiplication operator \(\gamma\) is allowed to change the sign in a sub-region \(P\) of the unit square \(Y\), is considered. Outside of \(P\) the coefficient is piecewise constant with pieces \(R\) (large constant) and \(H\) (constant 1). The study is motivated by issues associated with so-called electromagnetic metamaterials, where material properties encoded in \(\gamma\), which for classical materials would be strictly positive definite, are generalized to allow for sign changing \(\gamma\). The problem is approached by a (initially formal) power series ansatz for a solution and the eigenparameter with respect to an introduced oscillation parameter \(\eta\) producing a cascade of equations for the coefficient functions. The resulting equations involve a continuous linear operator \(T_{z}=\left(\nabla\cdot\nabla\right)^{-1}\nabla\cdot\gamma_{z}\nabla\), where \(\nabla\) acts on periodic functions with weak \(L^{2}\)-derivatives restricted to the subset \(Y\setminus\overline{R}=H\cup P\) of the unit square, which are perpendicular to \(1\), and \(-\nabla \cdot \) denoting the corresponding adjoint \(\nabla^{*}\). Here the coefficient \(\gamma_{z}\) is specified by \(\gamma_{z} := z \) for \(x\in P\) and \(1\) otherwise. The spectral properties of the operator \(T_{z}\), which for real \(z\) is selfadjoint with pure point spectrum, are studied in detail and a spectral representation is given. The analysis of \(T_{z}\) is linked to eigenvalue problems associated with the special case \(z=-1\). Based on an invertibility result for \(T_{z}\) solvability of the cascading system of equations can be established recursively. Finally, convergence of the power series ansatz is established.

MSC:
35Q60 PDEs in connection with optics and electromagnetic theory
35J70 Degenerate elliptic equations
78A48 Composite media; random media in optics and electromagnetic theory
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