The perturbed Maxwell operator as pseudodifferential operator (Q2439227)

The authors recall the \(L^2\)-theory of electromagnetism. The dynamical equations \[ \partial_tE=+ \varepsilon^{-1}\nabla_x\times H,\qquad \partial_tH=- \mu^{-1}\nabla_x\times E \] can be recast as a time-dependent Schrödinger equation \[ i\partial_t\Psi= M_w\Psi,\tag{2} \] where \(\Psi= (E,H)\) consist of the electric field \(E= (E_1,E_2,E_3)\) and the magnetic field \(H= (H_1,H_2,H_3)\) and \[ M_w= \begin{pmatrix} 0 & +i\varepsilon^{-1}\nabla^x_x\\ -i\mu^{-1}\nabla^x_x & 0\end{pmatrix} \] is the Maxwell operator. The symbol \(\nabla^x_x\) is shorthand for the curl. The second set of Maxwell equations which imposes the absence of sources \[ \nabla_x\cdot\varepsilon E= 0,\qquad \nabla_x\cdot\mu H= 0 \] enter as a constraint on the initial conditions for equation (2), or, equivalently one can restrict the domain to the physical states of \(M_w\). The authors assume that \(\varepsilon\), \(\mu\) in \(L^\infty(\mathbb{R}^3, \mathrm{Mat}_{\mathbb{C}}(3))\) are Hermitian-matrix-valued functions which are bounded away from \(0\) and \(+\infty\), i.e., \(0<c \mathrm{id}_{\mathbb{R}^3}\leq\varepsilon\), \(\mu\leq C \mathrm{id}_{\mathbb{R}^3}\) for \(0< c\leq C<+\infty\). The periodic Maxwell operator acting on the fiber at \(k\) is \[ M_0(k)= \begin{pmatrix} \varepsilon^{-1}(\widehat y) & 0\\ 0 &\mu^{-1}(\widehat y)\end{pmatrix} \begin{pmatrix} 0 & -(-i\nabla_y+ k)^x\\ +(-i\nabla_y+ k)^x & 0\end{pmatrix}. \] To derive effective dynamics and ray optics, the authors prove that the perturbed periodic Maxwell operator \(M_0\) in \(d=3\) can be seen as a pseudodifferential operator. They characterize the behavior of \(M_0\) and the physical initial states at small crystal momenta \(k\) and small frequencies. They prove that the band spectrum is generically symmetric with respect to inversions at \(k=0\) and that there are exactly four ground state bands with approximately linear dispersion near \(k=0\).