\begin{appendix}
	\section{Properties of $M_{EE}^2(k)$ and $M_{HH}^2(k)$} 
	\label{appendix:proof_properties_wave_operators}
	
	
	\begin{proof}[Lemma~\ref{periodic:lem:essential_properties_wave_operators}]
		Just like before, we will only spell out some of the proofs for the electric field operator $M_{EE}^2$, everything applies to $M_{HH}^2$ after exchanging the roles of $\eps$ and $\mu$. 
		
		\begin{enumerate}[(1)]
			\item This follows immediately from the selfadjointness of $M_{EE}^2$ and $M_{HH}^2$. 
			
			\item Because of the $\Gamma$-periodicity of $\eps$ and $\mu$, the fiber operators necessarily satisfy the equivariance condition~\eqref{periodic:eqn:equivariant_operator}. 
		
			To show analyticity, we note that $M_{EE}^2(k)$ is a quadratic polynomial in $k$, consisting of $M_{EE}^2(0)$ and lower-order operators which are infinitesimally small compared to it. Hence, we deduce from the Kato-Rellich theorem \cite[Theorem~X.12]{Reed_Simon:M_cap_Phi_2:1975} that the domain $\domain \bigl ( M_{EE}^2(k) \bigr ) = \domain \bigl ( M_{EE}^2(0) \bigr )$ is independent of $k$ as well. Combining these two factoids yields analyticity of $k \mapsto M_{EE}^2(k)$ for all $k \in \R^3$. 
			
			
			\item While we could show this directly, starting from the definitions of $M_{EE}^2$ and $M_{HH}^2$, we will give a proof that exploits the connection to $\Maux$ and what has already been proven about the auxiliary Maxwell operator in the literature. 
			
			To make the notation more compact, we abbreviate the divergence-free electromagnetic, electric and magnetic subspaces with
			
			\begin{align*}
				\mathcal{J}(k) :& \negmedspace = \ker \, \bigl ( \Div(k) \, W \bigr )
				\subset L^2_W(\T^3,\C^6)
				, 
				\\
				\mathcal{J}^E(k) :& \negmedspace = \ker \, \bigl ( (\nabla - \ii k) \cdot \eps \bigr )
				\subset L^2_{\eps}(\T^3,\C^3)
				, 
				\\
				\mathcal{J}^H(k) :& \negmedspace = \ker \, \bigl ( (\nabla - \ii k) \cdot \mu \bigr )
				\subset L^2_{\mu}(\T^3,\C^3)
				, 
			\end{align*}
			
			where $\Div(k) \bigl ( \psi^E , \psi^H \bigr ) := \bigl ( (\nabla - \ii k) \, \cdot \, \psi^E \, , \, (\nabla - \ii k) \, \cdot \, \psi^H \bigr )$ is the electromagnetic divergence in the fiber labeled with Bloch momentum $k$. 
			
			According to \cite[Theorem~3.4]{DeNittis_Lein:adiabatic_periodic_Maxwell_PsiDO:2013} the resolvent $\bigl ( \Maux(k) \, \vert_{\mathcal{J}(k)} - z \bigr )^{-1}$ is compact when $z \in \C \setminus \R$ as the spectrum of the selfadjoint operator $\Maux(k)$ is necessarily a subset of the reals. Then $z^2$ cannot be an element of $\sigma \bigl ( \Maux(k)^2 \bigr )$ and we therefore deduce that also 
			
			\begin{align*}
				\bigl ( \Maux(k)^2 \, \vert_{\mathcal{J}(k)} - z^2 \bigr )^{-1} &= \bigl ( \Maux(k) \, \vert_{\mathcal{J}(k)} - z \bigr )^{-1} \; \bigl ( \Maux(k) \, \vert_{\mathcal{J}(k)} + z \bigr )^{-1}
				\\
				&= \bigl ( M_{EE}^2(k) \, \vert_{\mathcal{J}^E(k)} - z^2 \bigr )^{-1} \oplus \bigl ( M_{HH}^2(k) \, \vert_{\mathcal{J}^H(k)} - z^2 \bigr )^{-1}
			\end{align*}
			
			is compact. Here, the direct sum refers to the decomposition of 
			
			\begin{align*}
				L^2_W(\T^3,\C^6) = L^2_{\eps}(\T^3,\C^3) \oplus L^2_{\mu}(\T^3,\C^3) 
			\end{align*}
			
			into electric and magnetic subspaces. This is evidently the case if and only if both, $\bigl ( M_{EE}^2(k) \, \vert_{\mathcal{J}^E(k)} - z^2 \bigr )^{-1}$ and $\bigl ( M_{HH}^2(k) \, \vert_{\mathcal{J}^H(k)} - z^2 \bigr )^{-1}$ are compact. 
		
			Hence, the non-negative operator $M_{EE}^2(k) \, \vert_{\mathcal{J}^E(k)}$ restricted to the transversal subspace and its magnetic counterpart $M_{HH}^2(k) \, \vert_{\mathcal{J}^H(k)}$ have purely discrete spectra, consisting solely of eigenvalues of finite multiplicity accumulating at $\infty$. 
			
			\item Proposition~\ref{setup:prop:identification_spaces}~(1) translates fiber-wise: $(\nabla - \ii k) \times (\nabla - \ii k) \varphi = 0$ holds for all $\varphi \in \Cont^{\infty}_{\mathrm{per}}(\T^3)$, and therefore gradient fields (which form an infinite-dimensional subspace) contribute to the essential spectrum at $0$. In fact, revisiting the arguments in the proof Proposition~\ref{setup:prop:identification_spaces}~(1), we deduce that the kernel of $M_{EE}^2(k)$ consists \emph{only} of gradient fields. 
		
			To show that the essential spectra consist only of $0$, we note that due to the Helmholtz-type decomposition 
			
			\begin{align*}
				L^2_{\eps}(\T^3,\C^6) = \ran \, (- \ii \nabla + k) \oplus \bigl ( \ran \, (- \ii \nabla + k) \bigr )^{\perp_{\eps}}
				= \ran \, (- \ii \nabla + k) \oplus \mathcal{J}^E(k) 
			\end{align*}
			
			for the electric field and the fact that the spectrum of the wave operator $M_{EE}^2(k) \, \vert_{\mathcal{J}^E(k)}$ restricted to the transversal subspace $\mathcal{J}^E(k)$ is purely discrete (point~(3)). That shows the claim. 
			
			
		\end{enumerate}
		
	\end{proof}
	
	
\end{appendix}
{"config": "arxiv", "file": "1806.07783/appendix.tex"}