text:
\section{Two equivalent mathematical descriptions of electromagnetism}
\label{setup}

Electromagnetic, electric and magnetic Chern numbers arise naturally, depending on the equation of motion one starts with. The purpose of this section is to introduce three equivalent equations which govern electromagnetic waves propagating in certain linear media. The properties of the medium are phenomenologically described by the electric permittivity $\eps \in L^{\infty} \bigl ( \R^3 \, , \, \mathrm{Mat}_{\C}(3) \bigr )$ and the magnetic permeability $\mu \in L^{\infty} \bigl ( \R^3 \, , \, \mathrm{Mat}_{\C}(3) \bigr )$; collectively, we will refer to

\begin{align*}
	W = \left (

	\begin{matrix}
		\eps & 0 \\
		0 & \mu \\
	\end{matrix}

	\right )
	\in L^{\infty} \bigl ( \R^3 \, , \, \mathrm{Mat}_{\C}(6) \bigr )
\end{align*}

as the \emph{material weights}, and throughout this article, we will impose the following

\begin{assumption}[Material weights]\label{setup:assumption:material_weights}

	\begin{enumerate}[(a)]
		\item The medium is \emph{lossless}, \ie $W(x) = W(x)^*$ takes values in the \emph{hermitian} matrices.
		\item The medium is \emph{not a negative index material}, \ie there exist positive constants $C \geq c > 0$ so that $c \, \id \leq W \leq C \, \id$ holds.
		\item The medium has no bianisotropy, \ie the block-offdiagonal terms of $W$ vanish.
	\end{enumerate}

\end{assumption}

Under these conditions (first-order) Maxwell's equations and the two (second-order) wave equations for electric and magnetic fields admit an $L^2$-theory. In particular, they give rise to selfadjoint operators acting on \emph{complex} electromagnetic, electric and magnetic fields, respectively. In the end, though, all three descriptions are equivalent (\cf Theorem~\ref{setup:thm:equivalence_frameworks}).

While all of this is standard, we would like to emphasize two important points before experts skip the remainder of this section:

\begin{enumerate}[(1)]
	\item To break time-reversal symmetry in a non-bianisotroptic medium — a prerequisite to have topological phenomena — the material weights $W \neq \overline{W}$ have to be complex (\cf \cite[Proposition~3.2 and Theorem~3.3]{DeNittis_Lein:symmetries_electromagnetism:2017}). In the context of this article, $W \neq \overline{W}$ is a necessary condition to have non-zero Chern numbers. For such media physically meaningful Maxwell equations have been derived only recently in \cite[Section~2]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}. Indeed, only the physically meaningful Maxwell equations are equivalent to either of the wave equations.

	\item The maps $\imath^{E,H}$ defined in equations~\eqref{setup:eqn:inclusion} below that enter as an auxiliary quantity in this section will play a pivotal rôle in the proof of our main result, Theorem~\ref{intro:thm:Chern_numbers_agree}.
\end{enumerate}

\subsection{First-order formalism: Maxwell's equations in matter}
\label{setup:first_order}

Under the above assumptions, we can give rigorous meaning to \emph{Maxwell's equations in matter}

\begin{subequations}\label{setup:eqn:Maxwell_equations}

	\begin{align}
		\left (

		\begin{matrix}
			\eps & 0 \\
			0 & \mu \\
		\end{matrix}

		\right ) \, \frac{\partial}{\partial t} \left (

		\begin{matrix}
			\psi^E(t) \\
			\psi^H(t) \\
		\end{matrix}

		\right ) &= \left (

		\begin{matrix}
			+ \nabla \times \psi^H(t) \\
			- \nabla \times \psi^E(t) \\
		\end{matrix}

		\right )
		&&
		\mbox{(dynamical equation)}
		\label{setup:eqn:Maxwell_equations:dynamics}
		\\
		\left (

		\begin{matrix}
			\nabla \cdot \eps \psi^E(t) \\
			\nabla \cdot \mu \psi^H(t) \\
		\end{matrix}

		\right ) &= \left (

		\begin{matrix}
			0 \\
			0 \\
		\end{matrix}

		\right )
		&&
		\mbox{(constraint equation)}
		\label{setup:eqn:Maxwell_equations:constraint}
		\\
		\left (

		\begin{matrix}
			\psi^E(t_0) \\
			\psi^H(t_0) \\
		\end{matrix}

		\right ) &= \left (

		\begin{matrix}
			\phi^E \\
			\phi^H \\
		\end{matrix}

		\right )
		&&
		\mbox{(initial condition)}
	\end{align}

\end{subequations}

defined on the vector space

\begin{align}
	\Hil = \Bigl \{ \Psi \in L^2(\R^3,\C^6) \; \; \big \vert \; \; \Psi \mbox{ is a $\omega \geq 0$ frequency wave} \Bigr \}
\end{align}

composed of \emph{complex} non-negative frequency waves; we will give a mathematically precise definition of $\Hil$ in equation~\eqref{setup:eqn:first_order_Helmholtz_splitting} below. For the benefit of the reader, we present enough details here to give rigorous meaning to equations~\eqref{setup:eqn:Maxwell_equations}; a detailed derivation with additional explanations can be found in \cite[Sections~2–3]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}.

\subsubsection{Representing real-valued electromagnetic waves as complex $\omega \geq 0$ waves}
\label{setup:first_order:representation_complex_waves}

The key idea of \cite{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017} is to write a real electromagnetic field $(\mathbf{E},\mathbf{H}) = \Psi_+ + \Psi_-$ as the sum of two \emph{complex} waves $\Psi_{\pm}$ composed solely of non-negative ($+$) and non-positive ($-$) frequencies. As sources are absent and electromagnetic fields must be transversal, there are no zero frequency fields contributing to $\Psi_{\pm}$. Hence, we will call $\Psi_{\pm}$ the positive/negative frequency contribution. To ensure that their sum is real, $\Psi_+$ and

\begin{align}
	\Psi_- = \overline{\Psi_+}
	\label{setup:eqn:phase_locking_constraint}
\end{align}

are phase locked. Put another way, $\Psi_+$ and $\Psi_-$ are \emph{not} independent degrees of freedom, and we may pick one of the two — typically $\Psi_+$ — to describe the real wave $(\mathbf{E},\mathbf{H}) = 2 \Re \Psi_+$. In this sense, $\Psi_+$ is a complex wave \emph{representing} the real electromagnetic field $(\mathbf{E},\mathbf{H})$. As we shall see below, this correspondence $(\mathbf{E},\mathbf{H}) \leftrightarrow \Psi_+$ is one-to-one (\cf Proposition~\ref{setup:prop:complex_representation_real_fields}).

When the weights $W \neq \overline{W}$ are complex, then $\Psi_+$ and $\Psi_-$ evolve according to \emph{different} Maxwell equations: in order to ensure the phase locking condition~\eqref{setup:eqn:phase_locking_constraint}, the Maxwell equations for $\Psi_-$ involve the complex conjugate weights. Thus, the restriction of \eqref{setup:eqn:Maxwell_equations} to non-negative frequencies is crucial as the negative frequency solutions to \eqref{setup:eqn:Maxwell_equations} \emph{sans} frequency restriction are unphysical.

One of the main points was to show that Maxwell's equations~\eqref{setup:eqn:Maxwell_equations} of lossless positive index media can be recast in the form of a Schrödinger equation

\begin{align}
	\ii \partial_t \Psi(t) = M \Psi(t)
	,
	&&
	\Psi(t_0) = \Phi
	.
	\label{setup:eqn:Schroedinger_form_Maxwell_equations}
\end{align}

To rigorously define the \emph{Maxwell operator} $M$, we first multiply both sides of \eqref{setup:eqn:Maxwell_equations:dynamics} with $\ii \, W^{-1}$, which yields $\ii \partial_t \Psi(t) = \Maux \, \Psi(t)$ where the \emph{auxiliary Maxwell operator}

\begin{align}
	\Maux := W^{-1} \, \Rot := \left (

	\begin{matrix}
		\eps^{-1} & 0 \\
		0 & \mu^{-1} \\
	\end{matrix}

	\right ) \, \left (

	\begin{matrix}
		0 & + \ii \nabla^{\times} \\
		- \ii \nabla^{\times} & 0 \\
	\end{matrix}

	\right )
	\label{setup:eqn:auxiliary_Maxwell_operator}
\end{align}

lacks the restriction to non-negative frequencies. Endowed with the domain $\domain(\Maux) := \domain(\Rot)$ of the free Maxwell operator (made explicit in \cite[equation~(15)]{DeNittis_Lein:adiabatic_periodic_Maxwell_PsiDO:2013}), $\Maux$ defines a closed operator on the \emph{Banach} space $L^2(\R^3,\C^6)$. Once we equip this $L^2$-space with the weighted \emph{energy scalar product}

\begin{align}
	\scpro{\Phi}{\Psi}_W := \bscpro{\Phi}{W \, \Psi}_{L^2(\R^3,\C^6)}
	,
	\label{setup:eqn:energy_scalar_product}
\end{align}

we obtain the Hilbert space $L^2_W(\R^3,\C^6)$; note that due to our assumptions on the material weights, $L^2_W(\R^3,\C^6)$ agrees with the ordinary, unweighted $L^2(\R^3,\C^6)$ as Banach spaces.

On this weighted $L^2$-space $\Maux = \bigl ( \Maux \bigr )^{\ast_W}$ is selfadjoint \cite[Proposition~6.2]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}, and the spectral projections

\begin{subequations}

	\begin{align}
		P_+ :& \negmedspace = 1_{(0,\infty)}(\Maux)
		\label{setup:eqn:positive_frequency_projection}
		\\
		P_0 :& \negmedspace = 1_{\{ 0 \}}(\Maux)
		\label{setup:eqn:zero_frequency_projection}
	\end{align}

\end{subequations}

onto the positive and zero frequency contributions can be defined via functional calculus. The Hilbert space of complex $\omega \geq 0$ waves is the corresponding spectral subspace,

\begin{align}
	\Hil :& \negmedspace = 1_{[0,\infty)}(\Maux) \bigl [ L^2_W(\R^3,\C^6) \bigr ]
	\notag \\
	&= \mathcal{J}_+ \oplus \mathcal{G}
	:= P_+ \bigl [ L^2_W(\R^3,\C^6) \bigr ] \oplus P_0 \bigl [ L^2_W(\R^3,\C^6) \bigr ]
	,
	\label{setup:eqn:first_order_Helmholtz_splitting}
\end{align}

that we further divide into positive and zero frequency components. This \emph{Helmholtz splitting} (\cf \cite[Section~3.2.1]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}) is conceptually important since longitudinal gradient fields that make up $\mathcal{G}$ are $\scpro{\, \cdot \,}{\, \cdot \,}_W$-orthogonal to the positive frequency subspace $\mathcal{J}_+$ — so that its elements therefore automatically satisfy the transversality constraint~\eqref{setup:eqn:Maxwell_equations:constraint} in the weak sense.

Any real electromagnetic field $(\mathbf{E},\mathbf{H}) \in L^2(\R^3,\R^6)$ has a unique representative $\Phi := Q (\mathbf{E},\mathbf{H})$ via the map

\begin{align}
	Q := \bigl ( P_+ + \tfrac{1}{2} P_0 \bigr ) \big \vert_{L^2(\R^3,\R^6)} : L^2(\R^3,\R^6) \longrightarrow \Hil
	.
	\label{setup:eqn:Q_complex_representative_map}
\end{align}

\begin{proposition}[{{\cite[Corollary~A.2]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}}}]\label{setup:prop:complex_representation_real_fields}
	Suppose the material weights satisfy Assumption~\ref{setup:assumption:material_weights}. Then the maps

	\begin{align*}
		Q &: L^2(\R^3,\R^6) \longrightarrow \Hil
		\\
		2 \Re \, Q &: L^2(\R^3,\R^6) \longrightarrow \Hil
	\end{align*}

	are injective. Hence, any real electromagnetic field $(\mathbf{E},\mathbf{H}) \in L^2(\R^3,\R^6)$ can be uniquely represented as a complex wave.
\end{proposition}

Morally speaking, this one-to-one correspondence can be understood as follows: complexifying the \emph{real} vector space $L^2(\R^3,\C^6)$ “doubles” the degrees of freedom. One way to eliminate the superfluous elements is to restrict to complex waves of non-negative frequencies. A more careful analysis shows \cite[Lemma~2.5]{DeNittis_Lein:ray_optics_photonic_crystals:2014} that for real transversal fields, the map $P_+$ restricted to the subspace of real transversal fields is really a bijection onto $\mathcal{J}_+$.

\begin{remark}
	When $W = \overline{W}$ is real, then a quick computation yields $Q \vert_{\ran Q}^{-1} = 2 \Re$ is just twice the real part. This computation also explains the presence of the factor $\nicefrac{1}{2}$ in euqation~\eqref{setup:eqn:Q_complex_representative_map}, it avoids $\omega = 0$ fields being counted twice. Even though we suspect this is also true for media with complex weights $W \neq \overline{W}$, we are presently not aware of a proof. For details we refer the interested reader to Section~3.2.2 and Appendix~A of \cite{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}.
\end{remark}

\subsubsection{The Schrödinger formalism of electromagnetism}
\label{setup:first_order:Schroedinger_formalism}

The analog of the quantum Hamiltonian which enters the Schrödinger-type equation~\eqref{setup:eqn:Schroedinger_form_Maxwell_equations} is the Maxwell operator $M := \Maux \vert_{\omega \geq 0}$, that is obtained by restricting the \emph{auxiliary} Maxwell operator to the non-negative frequency Hilbert space $\Hil$. Hence, endowed with the obvious domain $\domain(M) := \domain(\Maux) \cap \Hil$ the Maxwell operator $M = M^{\ast_W}$ inherits the selfadjointness from its parent \cite[Lemma~B.2]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}.

Then a straightforward analysis shows that Maxwell's equations~\eqref{setup:eqn:Maxwell_equations} are equivalent to the Schrödinger equation~\eqref{setup:eqn:Schroedinger_form_Maxwell_equations}.

\begin{theorem}[Equivalence of \eqref{setup:eqn:Maxwell_equations} and \eqref{setup:eqn:Schroedinger_form_Maxwell_equations} {{\cite[Theorem~3.4]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}}}]\label{setup:thm:equivalence_frameworks}
	Suppose the material weights $W$ which describe the medium satisfy Assumption~\ref{setup:assumption:material_weights}.
	Then the Maxwell equations~\eqref{setup:eqn:Maxwell_equations} are equivalent to the Schrödinger-type equation~\eqref{setup:eqn:Schroedinger_form_Maxwell_equations}. The real electromagnetic field $\bigl ( \mathbf{E}(t) , \mathbf{H}(t) \bigr ) = Q^{-1} \, \Psi(t)$ can be recovered from the inverse of the map $Q$.
\end{theorem}

\subsection{Second-order formalism: wave equations for electric and magnetic fields}
\label{setup:second_order}

Alternatively, we can square the Schrödinger-type equation~\eqref{setup:eqn:Schroedinger_form_Maxwell_equations} to obtain second-order wave equations; this has the advantage of yielding separate equations for electric

\begin{subequations}\label{setup:eqn:wave_equation_electric}

	\begin{align}
		&\partial_t^2 \psi^E(t) + \eps^{-1} \, \nabla^{\times} \, \mu^{-1} \, \nabla^{\times} \psi^E(t) = 0
		,
		&&
		\mbox{(dynamical equation)}
		\label{setup:eqn:wave_equation_electric:dynamics}
		\\
		&\nabla \cdot \eps \psi(t) = 0 = \nabla \cdot \eps \, \partial_t \psi(t)
		,
		&&
		\mbox{(constraint equation)}
		\label{setup:eqn:wave_equation_electric:constraint}
		\\
		&\psi^E(t_0) = \phi^E
		, \; \,
		\partial_t \psi^E(t_0) = + \eps^{-1} \, \nabla \times \phi^H
		,
		&&
		\mbox{(initial conditions)}
		\label{setup:eqn:wave_equation_electric:initial_conditions}
	\end{align}

\end{subequations}

and magnetic components,

\begin{subequations}\label{setup:eqn:wave_equation_magnetic}

	\begin{align}
		&\partial_t^2 \psi^H(t) + \mu^{-1} \, \nabla^{\times} \, \eps^{-1} \, \nabla^{\times} \psi^H(t) = 0
		,
		&&
		\mbox{(dynamical equation)}
		\label{setup:eqn:wave_equation_magnetic:dynamics}
		\\
		&\nabla \cdot \mu \psi(t) = 0 = \nabla \cdot \mu \, \partial_t \psi(t)
		,
		&&
		\mbox{(constraint equation)}
		\label{setup:eqn:wave_equation_magnetic:constraint}
		\\
		&\psi^H(t_0) = \phi^H
		, \; \,
		\partial_t \psi^H(t_0) = + \mu^{-1} \, \nabla \times \phi^H
		.
		&&
		\mbox{(initial conditions)}
		\label{setup:eqn:wave_equation_magnetic:initial_conditions}
	\end{align}

\end{subequations}

This is because the auxiliary Maxwell operator $\Maux = \left (

\begin{smallmatrix}
	0 & + \ii \eps^{-1} \nabla^{\times} \\
	- \ii \mu^{-1} \nabla^{\times} & 0 \\
\end{smallmatrix}

\right )$ is completely block-\emph{off}diagonal. Therefore, its square

\begin{align}
	(\Maux)^2 = \left (

	\begin{matrix}
		M_{EE}^2 & 0 \\
		0 & M_{HH}^2 \\
	\end{matrix}

	\right )
	:= \left (

	\begin{matrix}
		\eps^{-1} \, \nabla^{\times} \, \mu^{-1} \, \nabla^{\times} & 0 \\
		0 & \mu^{-1} \, \nabla^{\times} \, \eps^{-1} \, \nabla^{\times} \\
	\end{matrix}

	\right )
	\label{setup:eqn:squared_auxiliary_Maxwell_operator}
\end{align}

is block-\emph{diagonal}, and the operators in the block-diagonals are those we are interested in. In fact, this equation encapsulates their precise mathematical definitions: the domains of $M_{EE}^2$ and $M_{HH}^2$ are electric and magnetic part of the domain of $(\Maux)^2 = M_{EE}^2 \oplus M_{HH}^2$,

\begin{align*}
	\domain \bigl ( M_{EE}^2 \bigr ) :& \negmedspace = \domain \bigl ( (\Maux)^2 \bigr )^E
	:= \Bigl \{ \psi^E \in L^2_{\eps}(\R^3,\C^3) \; \; \big \vert \; \;
	\Psi = \left (

	\begin{smallmatrix}
		\psi^E \\
		\psi^H \\
	\end{smallmatrix}

	\right ) \in \domain \bigl ( (\Maux)^2 \bigr )
	\Bigr \}
	,
	\\
	\domain \bigl ( M_{HH}^2 \bigr ) :& \negmedspace = \domain \bigl ( (\Maux)^2 \bigr )^H
	:= \Bigl \{ \psi^H \in L^2_{\mu}(\R^3,\C^3) \; \; \big \vert \; \;
	\Psi = \left (

	\begin{smallmatrix}
		\psi^E \\
		\psi^H \\
	\end{smallmatrix}

	\right ) \in \domain \bigl ( (\Maux)^2 \bigr )
	\Bigr \}
	.
\end{align*}

\begin{remark}
	This definition via $\Maux$ imposes only the bare minimum of conditions on the domain; in principle, we \emph{could} attempt to define $M_{EE}^2$ and $M_{HH}^2$ directly without making any reference to the first-order operators, but that is actually more delicate if $\eps$ and $\mu$ are not $\Cont^1$-regular with bounded first-order derivatives.
\end{remark}

Consequently, $M_{EE}^2$ and $M_{HH}^2$, seen as operators on the electric Hilbert space $L^2_{\eps}(\R^3,\C^3)$ and the magnetic Hilbert space $L^2_{\mu}(\R^3,\C^3)$, are selfadjoint. These Hilbert spaces are defined just as $L^2_W(\R^3,\C^6)$ by endowing the \emph{Banach space} $L^2(\R^3,\C^3)$ with the weighted scalar products

\begin{align*}
	\bscpro{\phi^E}{\psi^E}_{\eps} :& \negmedspace = \bscpro{\phi^E}{\eps \psi^E}_{L^2(\R^3,\C^3)}
	,
	\\
	\bscpro{\phi^H}{\psi^H}_{\mu} :& \negmedspace = \bscpro{\phi^H}{\mu \psi^H}_{L^2(\R^3,\C^3)}
	.
\end{align*}

\begin{lemma}
	$M_{EE}^2 = (M_{EE}^2)^{\ast_{\eps}}$ and $M_{HH}^2 = (M_{HH}^2)^{\ast_{\mu}}$ are selfadjoint.
\end{lemma}

While equations~\eqref{setup:eqn:wave_equation_electric} and \eqref{setup:eqn:wave_equation_magnetic} makes it seem as if electric and magnetic fields decouple, this is of course not the case: as second-order equations, we not only need to specify $\psi^E(t_0)$ or $\psi^H(t_0)$ but also the time-derivative

\begin{subequations}\label{setup:eqn:time_derivative_intial_conditions_second_order}

	\begin{align}
		\partial_t \psi^E(t_0) &= + \eps^{-1} \, \nabla \times \psi^H(t_0)
		\label{setup:eqn:time_derivative_intial_conditions_second_order:electric}
		\\
		\partial_t \psi^H(t_0) &= - \mu^{-1} \, \nabla \times \psi^E(t_0)
		\label{setup:eqn:time_derivative_intial_conditions_second_order:magnetic}
	\end{align}

\end{subequations}

that evidently has to satisfy Maxwell's equations~\eqref{setup:eqn:Maxwell_equations}. Part and parcel is the assumption that $\Psi(t_0) = \bigl ( \psi^E(t_0) , \psi^H(t_0) \bigr ) \in \Hil$ is composed of positive frequencies.

Lastly, akin to \eqref{setup:eqn:first_order_Helmholtz_splitting} let us introduce the Helmholtz decomposition for the electric and magnetic Hilbert spaces,

\begin{subequations}\label{setup:eqn:second_order_Helmholtz_splitting}

	\begin{align}
		L^2_{\eps}(\R^3,\C^3) &= \bigl ( \ran \nabla \bigr )^{\perp_{\eps}} \oplus \bigl ( \ran \nabla \bigr )
		= \ker \, ( \nabla \cdot \eps ) \oplus \bigl ( \ran \nabla \bigr )
		,
		\label{setup:eqn:second_order_Helmholtz_splitting:electric}
		\\
		L^2_{\mu}(\R^3,\C^3) &= \bigl ( \ran \nabla \bigr )^{\perp_{\mu}} \oplus \bigl ( \ran \nabla \bigr )
		= \ker \, ( \nabla \cdot \mu ) \oplus \bigl ( \ran \nabla \bigr )
		.
		\label{setup:eqn:second_order_Helmholtz_splitting:magnetic}
	\end{align}

\end{subequations}

Here, the transversal fields are those that are $\scpro{\, \cdot \,}{\, \cdot \,}_{\eps}$- and $\scpro{\, \cdot \,}{\, \cdot \,}_{\mu}$-orthogonal to the gradient fields. For a rigorous definition of the gradient $\nabla$, the curl $\nabla^{\times}$ and the divergence $\nabla \cdot$ we refer to \cite[Appendix~A]{DeNittis_Lein:adiabatic_periodic_Maxwell_PsiDO:2013}.

\subsection{Equivalence of first- and second-order equations}
\label{setup:equivalence}

Put as a mathematical statement, the equivalence reads:

\begin{theorem}[Equivalence of first- and second-order formalism]\label{setup:thm:equivalence}
	Suppose the medium is described by material weights that satisfy Assumption~\ref{setup:assumption:material_weights}, and the (complex) initial state $\Phi = (\phi^E , \phi^H) \in \Hil$ satisfies the constraint equation~\eqref{setup:eqn:Maxwell_equations:constraint}.

	\begin{enumerate}[(1)]
		\item Equations~\eqref{setup:eqn:Maxwell_equations}, \eqref{setup:eqn:wave_equation_electric} and \eqref{setup:eqn:wave_equation_magnetic} are all equivalent in the following sense: electric and magnetic part of the solution $\Psi(t) = \bigl ( \psi^E(t) , \psi^H(t) \bigr )$ to the first-order Maxwell equations~\eqref{setup:eqn:Maxwell_equations} satisfy the wave equations~\eqref{setup:eqn:wave_equation_electric} and \eqref{setup:eqn:wave_equation_magnetic}, respectively. Conversely, we can reconstruct \emph{the} solution to Maxwell's equations~\eqref{setup:eqn:Maxwell_equations} from solution to the electric or magnetic wave equation alone. And hence, we can obtain \emph{the} solution to the magnetic wave equation~\eqref{setup:eqn:wave_equation_magnetic} from that of the electric wave equation~\eqref{setup:eqn:wave_equation_electric} and vice versa.

		\item The real-valued electromagnetic field $\bigl ( \mathbf{E}(t) , \mathbf{H}(t) \bigr )$ is represented by the complex solution $\Psi(t) = Q \bigl ( \mathbf{E}(t) , \mathbf{H}(t) \bigr )$ with the help of the injective map $Q : L^2(\R^3,\R^6) \longrightarrow \Hil$ defined through equation~\eqref{setup:eqn:Q_complex_representative_map}.
	\end{enumerate}

\end{theorem}

When the material weights $W = \overline{W}$ are \emph{real}, this has long been known (see \eg \cite{Wilcox:scattering_theory_classical_physics:1966,Reed_Simon:scattering_theory_wave_equations:1977}). However, for \emph{complex} weights $W \neq \overline{W}$ this is new. Interestingly, the hurdle for an extension to complex material weights was to find \emph{physically meaningful first-order} Maxwell equations~\eqref{setup:eqn:Maxwell_equations} that are compatible with the real-valuedness of the physical fields $(\mathbf{E},\mathbf{H})$. Their derivation was one of the main aims of a recent work of ours \cite[Section~2]{DeNittis_Lein:Schroedinger_formalism_classical_waves:2017}, and this work is as an addendum.

We emphasize that Theorem~\ref{setup:thm:equivalence} applies to media with real material weights as well: while we could equivalently work with real electric fields $\mathbf{E}$ \emph{directly} in equations~\eqref{setup:eqn:wave_equation_electric} and \eqref{setup:eqn:Maxwell_equations}, the alternate strategy to represent $\mathbf{E} = 2 \Re \psi^E$ as a complex $\omega \geq 0$ wave works just as well and has the added advantage of extending to complex $\eps$ and $\mu$.

The proof consists of two parts, and only the first step requires a bit of work: firstly, we need to establish that our definition of $M_{EE}^2$ and $M_{HH}^2$ via the \emph{auxiliary} Maxwell operator (that lacks any frequency restriction) is compatible with the frequency constraint. The second step consists of showing a connection between the second initial condition $\partial_t \psi^E(t_0)$ for the second-order equation and $\psi^H(t_0)$ for the electric field wave equation~\eqref{setup:eqn:wave_equation_electric}, and an analogous statement for \eqref{setup:eqn:wave_equation_magnetic}.

\subsubsection{Equations~\eqref{setup:eqn:wave_equation_electric} and \eqref{setup:eqn:wave_equation_magnetic}, and the $\omega \geq 0$ frequency constraint}
\label{setup:equivalence:spaces}

The main ingredient in the proof of the first step is the map

\begin{subequations}\label{setup:eqn:inclusion}

	\begin{align}
		\imath^E &: \ker \, ( \nabla \cdot \eps ) \longrightarrow \mathcal{J}_+
		,
		\qquad
		\psi^E \mapsto \left (

		\begin{matrix}
			\psi^E \\
			- \ii \mu^{-1} \, \nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \psi^E \\
		\end{matrix}

		\right )
		,
		\label{setup:eqn:inclusion:electric}
		\\
		\imath^H &: \ker \, ( \nabla \cdot \mu ) \longrightarrow \mathcal{J}_+
		,
		\qquad
		\psi^H \mapsto \left (

		\begin{matrix}
			+ \ii \eps^{-1} \, \nabla^{\times} \bigl ( M_{HH}^2 \bigr )^{- \nicefrac{1}{2}} \psi^H \\
			\psi^H \\
		\end{matrix}

		\right )
		,
		\label{setup:eqn:inclusion:magnetic}
	\end{align}

\end{subequations}

for the electric field and for the magnetic field. Figotin and Klein gave the explicit map to reconstruct the electric part from the magnetic part as \cite[equations~(10)–(11)]{Figotin_Klein:localization_classical_waves_II:1997}, although they assumed that $\eps$ and $\mu$ are real and scalar-valued; the difference in sign in \cite[equations~(10)]{Figotin_Klein:localization_classical_waves_II:1997} stems from their choice to use $(\mathbf{H},\mathbf{E})$ rather than $(\mathbf{E},\mathbf{H})$ as the electromagnetic field (\cf \cite[equation~(4)]{Figotin_Klein:localization_classical_waves_II:1997}).

For the sake of concreteness, let us focus on the electric field. It turns out that $\imath^E$ is a bounded injection that maps transversal electric fields onto transversal, positive frequency \emph{electromagnetic} fields; its left-inverse $\mathrm{pr}^E : \Psi = \left (

\begin{smallmatrix}
	\psi^E \\
	\psi^H \\
\end{smallmatrix}

\right ) \mapsto \psi^E$ discards the magnetic component.

\begin{proposition}\label{setup:prop:identification_spaces}
	Suppose Assumption~\ref{setup:assumption:material_weights} on the weights holds.

	\begin{enumerate}[(1)]
		\item $\ker M_{EE}^2 = \ran \nabla = \ker M_{HH}^2$ where $\nabla$ is seen as an operator $\domain(\nabla) \subset L^2(\R^3,\C) \longrightarrow L^2(\R^3,\C^3)$.

		\item The electric and magnetic parts of $\mathcal{J}_+$ coincide with the divergence-free electric and magnetic fields in the sense that

		\begin{align*}
			L^2_{\eps}(\R^3,\C^3) \supset
			\ker \, ( \nabla \cdot \eps ) &= \mathcal{J}_+^E
			:= \Bigl \{ \psi^E \in L^2_{\eps}(\R^3,\C^3) \; \; \big \vert \; \;
			\Psi = \left (

			\begin{smallmatrix}
				\psi^E \\
				\psi^H \\
			\end{smallmatrix}

			\right ) \in \mathcal{J}_+
			\Bigr \}
			,
			\\
			L^2_{\mu}(\R^3,\C^3) \supset
			\ker \, ( \nabla \cdot \mu ) &= \mathcal{J}_+^H
			:= \Bigl \{ \psi^H \in L^2_{\mu}(\R^3,\C^3) \; \; \big \vert \; \;
			\Psi = \left (

			\begin{smallmatrix}
				\psi^E \\
				\psi^H \\
			\end{smallmatrix}

			\right ) \in \mathcal{J}_+
			\Bigr \}
			.
		\end{align*}

		\item The electric and magnetic components

		\begin{align*}
			\Hil^E :& \negmedspace = \Bigl \{ \psi^E \in L^2_{\eps}(\R^3,\C^3) \; \; \big \vert \; \;
			\Psi = \left (

			\begin{smallmatrix}
				\psi^E \\
				\psi^H \\
			\end{smallmatrix}

			\right ) \in \Hil
			\Bigr \}
			= L^2_{\eps}(\R^3,\C^3)
			,
			\\
			\Hil^H :& \negmedspace = \Bigl \{ \psi^H \in L^2_{\mu}(\R^3,\C^3) \; \; \big \vert \; \;
			\Psi = \left (

			\begin{smallmatrix}
				\psi^E \\
				\psi^H \\
			\end{smallmatrix}

			\right ) \in \Hil
			\Bigr \}
			= L^2_{\mu}(\R^3,\C^3)
			,
		\end{align*}

		of the non-negative frequency space $\Hil$ coincide with $L^2_{\eps}(\R^3,\C^3)$ and $L^2_{\mu}(\R^3,\C^3)$.
	\end{enumerate}

\end{proposition}

This Proposition tells us several things: on a conceptual level it states that the subspace of transversal electric fields $\ker \, ( \nabla \cdot \eps )$ does not contain (unphysical) waves that cannot be mapped to a real, transversal electromagnetic field $(\mathbf{E},\mathbf{H}) \in L^2(\R^3,\R^6)$. Indeed, concatenating $Q$ with the projection $\mathrm{pr}^E$ gives us an \emph{injective} map $\mathrm{pr}^E \circ Q : L^2(\R^3,\R^6) \longrightarrow L^2_{\eps}(\R^3,\C^3)$.

And on a practical level it tells us that we could have defined $M_{EE}^2$ and $M_{HH}^2$ via $M^2$ rather than $(\Maux)^2$. Note, however, that $M^2 \neq M_{EE}^2 \oplus M_{HH}^2$ is not the direct sum of two operators because the Hilbert space $M$ is defined on does not — the $\omega \geq 0$ condition imposes a relation on electric and magnetic components.

\begin{proof}
	Since the roles of electric and magnetic fields are symmetric, it suffices to make the arguments explicit only for the electric field.

	\begin{enumerate}[(1)]
		\item We note that $M_{EE}^2 \psi^E = \eps^{-1} \, \nabla^{\times} \, \mu^{-1} \, \nabla^{\times} \psi^E = 0$ implies that either $\nabla^{\times} \psi^E = 0$ or $\mu^{-1} \, \nabla^{\times} \psi^E \in \ran \nabla = \ker \nabla^{\times}$ (as $\eps^{-1}$ and $\mu^{-1}$ are bounded and have bounded inverses). Thanks to $\nabla \times \nabla f = 0$, we deduce the inclusion $\ran \nabla \subseteq \ker M_{EE}^2$.

		In the second case we are looking for a vector that lies in the intersection

		\begin{align*}
			\ran \bigl ( \mu^{-1} \, \nabla^{\times} \bigr ) \cap \ran \nabla = \{ 0 \}
			,
		\end{align*}

		and we will show it is necessarily zero: the adapted Helmholtz decomposition \cite[Appendix~A]{DeNittis_Lein:adiabatic_periodic_Maxwell_PsiDO:2013} of $L^2_{\mu}(\R^3,\C^3) = \ran \nabla \oplus \bigl ( \ran \nabla \bigr )^{\perp_{\mu}}$ implies

		\begin{align*}
			\ran \bigl ( \mu^{-1} \, \nabla^{\times} \bigr ) &= \ker \, ( \nabla \cdot \mu )
			= \bigl ( \ran \nabla \bigr )^{\perp_{\mu}}
			\subset L^2_{\mu}(\R^3,\C^3)
			.
		\end{align*}

		Because $L^2_{\mu}(\R^3,\C^3) = L^2(\R^3,\C^3) = L^2_{\eps}(\R^3,\C^3)$ all coincide as \emph{Banach} spaces by our assumptions on $\eps$ and $\mu$, we conclude $\ran \bigl ( \mu^{-1} \, \nabla^{\times} \bigr ) \cap \ran \nabla = \{ 0 \}$ independently of the choice of scalar product. This shows the opposite inclusion $\ran \nabla \supseteq \ker M_{EE}^2$, and in combination yields $\ran \nabla = \ker M_{EE}^2$.

		\item By the Helmholtz composition $L^2_{\eps}(\R^3,\C^3) = \ran \nabla \oplus \ker \, ( \nabla \cdot \eps ) = \ker M_{EE}^2 \oplus \ker \, ( \nabla \cdot \eps )$ holds, and since $M_{EE}^2 \geq 0$ is non-negative, we can express the second summand as

		\begin{align*}
			\ker \, ( \nabla \cdot \eps ) = 1_{(0,\infty)} \bigl ( M_{EE}^2 \bigr ) \bigl [ L^2_{\eps}(\R^3,\C^3) \bigr ]
			.
		\end{align*}

		The same reasoning applies to $\mathcal{J}_+ = 1_{(0,\infty)}(\Maux) \bigl [ L^2_W(\R^3,\C^6) \bigr ]$: the gradient fields $\mathcal{G} = 1_{\{ 0 \}}(\Maux) \bigl [ L^2_W(\R^3,\C^6) \bigr ]$ (this time seen as a subspace of $L^2_W(\R^3,\C^6)$) make up the kernel of $\Maux$ and spectral calculus gives us for free that $\mathcal{J}_+$ is $\scpro{\, \cdot \,}{\, \cdot \,}_W$-orthogonal to the gradient fields. Manually writing out the orthogonality condition then yields that elements of $\mathcal{J}_+ \subset \ker \, (\Div \, W) = \mathcal{G}^{\perp_W} \subset L^2(\R^3,\C^6)$ satisfy the divergence-free condition $\Div \, W \, \Psi = 0$, which we can write out as $\nabla \cdot \eps \, \psi^E = 0 = \nabla \cdot \mu \, \psi^H$. This shows the inclusion $\mathcal{J}_+^E \subseteq \ker \, ( \nabla \cdot \eps )$.

		The other inclusion $\mathcal{J}_+^E \supseteq \ker \, ( \nabla \cdot \eps )$ requires a bit more work. The key here is the map $\imath^E$ defined by \eqref{setup:eqn:inclusion:electric} that associates to each divergence-free electric field an element in $\mathcal{J}_+$. Of course, we need to show that this map is well-defined as a map $\ker \, ( \nabla \cdot \eps ) \longrightarrow L^2_W(\R^3,\C^6)$ and then prove that it maps divergence-free electric fields onto \emph{positive} frequency electromagnetic fields (which are automatically divergence-free).

		To show that $\mu^{-1} \, \nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \psi^E$ defines an element of $L^2_{\mu}(\R^3,\C^3)$, we note that the squared auxiliary Maxwell operator

		\begin{align*}
			(\Maux)^2 \big \vert_{\ker \, (\Div \, W)} = M_{EE}^2 \big \vert_{\ker \, ( \nabla \cdot \eps )} \oplus M_{HH}^2 \big \vert_{\ker \, ( \nabla \cdot \mu )}
		\end{align*}

		restricted to divergence-free fields (including negative frequency waves) coincides with the restriction of $M_{EE}^2$ and $M_{HH}^2$ to the divergence-free electric and magnetic fields. Then for all divergence-free electromagnetic fields $\Psi = (\psi^E , \psi^H) \in \ker \, (\Div \, W)$ the norm of the operator

		\begin{align}
			\mathrm{sgn} \bigl ( \Maux \bigr ) \, \Psi &= \Maux \, \bigl ( (\Maux)^2 \bigr )^{-\nicefrac{1}{2}} \Psi
			= \left (

			\begin{matrix}
				+ \ii \eps^{-1} \, \nabla^{\times} \, \bigl ( M_{HH}^2 \bigr )^{- \nicefrac{1}{2}} \psi^H \\
				- \ii \mu^{-1} \, \nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \psi^E \\
			\end{matrix}

			\right )
			\label{setup:eqn:sgn_M_EE_2_equation}
			\\
			&= \bigl ( (\Maux)^2 \bigr )^{-\nicefrac{1}{2}} \, \Maux \Psi
			= \left (

			\begin{matrix}
				+ \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \, \ii \eps^{-1} \, \nabla^{\times} \, \psi^H \\
				- \bigl ( M_{HH}^2 \bigr )^{- \nicefrac{1}{2}} \, \ii \mu^{-1} \, \nabla^{\times} \psi^E \\
			\end{matrix}

			\right )
			\label{setup:eqn:sgn_M_EE_2_equation_alternate}
		\end{align}

		is necessarily bounded by $1 \cdot \snorm{\Psi}_W$: to justify \eqref{setup:eqn:sgn_M_EE_2_equation} we note that $\abs{\Maux} = \bigl ( (\Maux)^2 \bigr )^{\nicefrac{1}{2}}$ maps $\domain(\Maux) \cap \ker \, \bigl ( \Div \, W \bigr )$ onto $\ran \Maux = \ker \, \bigl ( \Div \, W \bigr )$, so that its inverse

		\begin{align*}
			\bigl ( (\Maux)^2 \bigr )^{- \nicefrac{1}{2}} = \babs{\Maux}^{-1} : \ker \, \bigl ( \Div \, W \bigr ) \longrightarrow \domain(\Maux) \cap \ker \, \bigl ( \Div \, W \bigr )
		\end{align*}

		maps divergence-free fields onto divergence-free fields from the domain. Hence, applying $\Maux$ from the left gives us a \emph{bounded} operator on $\ker \, \bigl ( \Div \, W \bigr )$.

		Therefore, $\nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} : \ker \, ( \nabla \cdot \eps ) \longrightarrow \ran \nabla^{\times} = \ker \, ( \nabla \cdot )$ maps to the divergence-free fields, so if we multiply from the left with $\mu^{-1}$, we indeed get something in $\ker \, (\nabla \cdot \mu)$. Moreover, we deduce that this is in fact a \emph{bounded} operator, and $\imath^E$ is well-defined as a map

		\begin{align*}
			\ker \, ( \nabla \cdot \eps ) \longrightarrow \ker \, (\Div \, W) = \ker \, ( \nabla \cdot \eps ) \oplus \ker \, ( \nabla \cdot \mu )
			.
		\end{align*}

		All that is left is to prove that $\imath^E(\psi^E)$ is a positive frequency wave. The reason we defined $\imath^E$ the way we did is readily apparent when we compare it with \eqref{setup:eqn:sgn_M_EE_2_equation}: the operator $\mathrm{sgn} \bigl ( \Maux \bigr ) \big \vert_{\ker \, (\Div \, W)}$ has two eigenvalues, $\pm 1$, and the positive frequency fields $\Psi \in \mathcal{J}_+$ form the eigenspace of $\mathrm{sgn} \bigl ( \Maux \big \vert_{\ker \, (\Div \, W)} \bigr )$ to the eigenvalue $+1$. By design $\imath^E$ gives us an eigenvector to the eigenvalue $+1$: because we can write $\mathrm{sgn}(\Maux)$ in both ways, \eqref{setup:eqn:sgn_M_EE_2_equation} and \eqref{setup:eqn:sgn_M_EE_2_equation_alternate}, we deduce

		\begin{align*}
			\ii \eps^{-1} \, \nabla^{\times} \, \bigl ( M_{HH}^2 \bigr )^{- \nicefrac{1}{2}} \psi^H = \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \, \ii \eps^{-1} \, \nabla^{\times} \, \psi^H
		\end{align*}

		and a similar equation for the other component. The important fact here is that $M_{HH}^2$ becomes $M_{EE}^2$ when we “commute” it with $+ \ii \eps^{-1} \, \nabla^{\times}$, which is what we will exploit when applying $\mathrm{sgn}(\Maux)$ to $\imath^E(\psi^E)$,

		\begin{align*}
			\mathrm{sgn}(\Maux) \, \imath^E(\psi^E) &= \left (

			\begin{matrix}
				\eps^{-1} \, \nabla^{\times} \, \bigl ( M_{HH}^2 \bigr )^{- \nicefrac{1}{2}} \mu^{-1} \, \nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \, \psi^E \\
				- \ii \mu^{-1} \, \nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \psi^E \\
			\end{matrix}

			\right )
			\\
			&= (+1) \, \left (

			\begin{matrix}
				\eps^{-1} \, \nabla^{\times} \, \mu^{-1} \, \nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{-1} \psi^E \\
				- \ii \mu^{-1} \, \nabla^{\times} \, \bigl ( M_{EE}^2 \bigr )^{- \nicefrac{1}{2}} \psi^E \\
			\end{matrix}

			\right )
			= + \imath^E(\psi^E)
			.
		\end{align*}

		Note that we may either use $\psi^E$ or $\psi^H$ as an independent variable, and choosing $\psi^H$ instead of $\psi^E$ gives us $\imath^H$.

		Evidently, its inverse — the projection $\mathrm{pr}^E : (\psi^E , \psi^H) \mapsto \psi^E$ onto the first component is bounded and a left-inverse to $\imath^E$, \ie we have $\mathrm{pr}^E \circ \imath^E = \id_{\ker \, ( \nabla \cdot \eps )}$. This shows the opposite inclusion, $\mathcal{J}_+^E \supseteq \ker \, ( \nabla \cdot \eps )$.

		\item This follows from the Helmholtz decompositions~\eqref{setup:eqn:second_order_Helmholtz_splitting} combined with (1) and (2).
	\end{enumerate}

\end{proof}

\begin{remark}
	A second way to verify that $\imath^E(\psi^E)$ is composed solely of positive frequencies relies on functional calculus for $M_{EE}^2 = \eps^{-1} \, \nabla^{\times} \, \mu^{-1} \nabla^{\times}$. If we assume for a moment that $\imath^E(\psi^E) \in \mathcal{D}(\Maux)$, then the following expressions are all well-defined:

	\begin{align*}
		\Maux \, \imath^E(\psi^E)
		&= \int_{(0,\infty)} \Maux \, \left (

		\begin{matrix}
			\dd 1_{\lambda} \bigl ( M_{EE}^2 \bigr ) \, \psi^E \\
			- \ii \lambda^{-\nicefrac{1}{2}} \, \mu^{-1} \, \nabla^{\times} \, \dd 1_{\lambda} \bigl ( M_{EE}^2 \bigr ) \, \psi^E \\
		\end{matrix}

		\right )
		\\
		&
		= \int_{(0,\infty)} \left (

		\begin{matrix}
			\lambda^{-\nicefrac{1}{2}} \, \eps^{-1} \, \nabla^{\times} \, \mu^{-1} \nabla^{\times} \, \dd 1_{\lambda} \bigl ( M_{EE}^2 \bigr ) \, \psi^E \\
			- \ii \, \mu^{-1} \, \nabla^{\times} \, \dd 1_{\lambda} \bigl ( M_{EE}^2 \bigr ) \, \psi^E \\
		\end{matrix}

		\right )
		\\
		&= \int_{(0,\infty)} \bigl ( + \lambda^{\nicefrac{1}{2}} \bigr ) \, \left (

		\begin{matrix}
			\dd 1_{\lambda} \bigl ( M_{EE}^2 \bigr ) \, \psi^E \\
			- \ii \lambda^{-\nicefrac{1}{2}} \, \mu^{-1} \, \nabla^{\times} \, \dd 1_{\lambda} \bigl ( M_{EE}^2 \bigr ) \, \psi^E \\
		\end{matrix}

		\right )
	\end{align*}

	Of course, if $\imath^E(\psi^E) \not \in \mathcal{D}(\Maux)$, we need to regularize: if we replace $\imath^E(\psi^E)$ with the cut off wave $\Psi_b := \imath^E \Bigl ( 1_{(0,b)} \bigl ( M_{EE}^2 \bigr ) \, \psi^E \Bigr )$ for $b < \infty$, the above computation then shows that for $\Psi_b$ only frequencies in the range $(0,\sqrt{b})$ are excited, negative frequencies are excluded. The limit $\Maux \Psi_b$ as $b \rightarrow \infty$ will in general not exist in $L^2_W(\R^3,\C^6)$, but nevertheless, we still see that $\imath^E(\psi^E)$ is composed solely of positive frequencies.
\end{remark}

When we showed part (2), we have indeed also furnished a proof for the following Corollary that will be useful for our discussion of periodic electromagnetic media in Section~\ref{periodic}:

\begin{corollary}\label{setup:cor:inclusion_bounded_injections}
	The maps $\imath^E : \ker \, ( \nabla \cdot \eps ) \longrightarrow \mathcal{J}_+$ and $\imath^H : \ker \, ( \nabla \cdot \mu ) \longrightarrow \mathcal{J}_+$ from equations~\eqref{setup:eqn:inclusion} are bounded injections with left-inverses $\mathrm{pr}^{E,H} : (\psi^E , \psi^H) \mapsto \psi^{E,H}$.
\end{corollary}

Another way the equivalence of electric and magnetic field equations manifests itself is as a \emph{unitary} equivalence of the positive frequency parts of $M_{EE}^2$ and $M_{HH}^2$.

\begin{corollary}
	The positive frequency part $M_{HH}^2 \vert_{\omega > 0} := M_{HH}^2 \, \big \vert_{\ker \, ( \nabla \cdot \mu )}$ is related to the positive frequency part of

	\begin{align*}
		M_{EE}^2 \vert_{\omega > 0} = U_{EH} \, M_{HH}^2 \vert_{\omega > 0} \, U_{EH}^{-1}
	\end{align*}

	via the \emph{unitary} map

	\begin{align*}
		U_{EH} := \mathrm{pr}^E \circ \imath^H

		: \ker \, ( \nabla \cdot \mu ) \subset L^2_{\mu}(\R^3,\C^3) \longrightarrow \ker \, ( \nabla \cdot \eps ) \subset L^2_{\eps}(\R^3,\C^3)
		.
	\end{align*}

\end{corollary}

This was already recognized by Figotin and Klein (\cf \cite[equations~(9)–(10)]{Figotin_Klein:localization_classical_waves_II:1997}), although they did not give a proof showing that $U_{EH}$ is well-defined.

\begin{proof}
	The well-definedness and invertibility of $U_{EH}$ are direct consequences of Proposition~\ref{setup:prop:identification_spaces}. Therefore unitarity follows from checking

	\begin{align*}
		\bscpro{U_{EH} \varphi^H}{U_{EH} \psi^H}_{\eps} &= \Bscpro{\mu^{-1} \, \nabla^{\times} \, \eps^{-1} \, \nabla^{\times} \, (M_{HH}^2)^{-\nicefrac{1}{2}} \varphi^H}{(M_{HH}^2)^{-\nicefrac{1}{2}} \psi^H}_{\mu}
		\\
		&= \Bscpro{\varphi^H}{(M_{HH}^2)^{+\nicefrac{1}{2}} \, (M_{HH}^2)^{-\nicefrac{1}{2}} \psi^H}_{\mu}
		= \scpro{\varphi^H}{\psi^H}_{\mu}
	\end{align*}

	by direct computation on the dense subset $\domain(M_{HH}^2) \cap \ker \, ( \nabla \cdot \mu )$, and extending this by density to all of $\ker \, ( \nabla \cdot \mu )$.
\end{proof}

\subsubsection{Equivalence of the dynamics}
\label{setup:equivalence:dynamics}

The actual proof of equivalence is completely standard, indeed the hard part was in properly defining Maxwell's equations and verify that the spaces on which the second-order equations are defined are correct.

\begin{proof}[Theorem~\ref{setup:thm:equivalence}]
	As before, we will only formulate the proof for the electric field. Moreover, to simplify the presentation we impose in addition that the complex initial condition $\Phi := Q (\mathbf{E}_0 , \mathbf{H}_0) \in \mathcal{D}(M^2) \cap \mathcal{J}_+$ holds. This just allows us to write out the proof for strong rather than weak solutions.

	Thanks to Theorem~\ref{setup:thm:equivalence_frameworks} the (strong) solution $\Psi(t) = \e^{- \ii (t - t_0) M} \Phi$ to Maxwell's equations~\eqref{setup:eqn:Maxwell_equations} can be expressed in terms of the Maxwell operator. Seeing as $\e^{- \ii t M} : \domain(M^2) \longrightarrow \domain(M^2)$ preserves the core $\domain(M^2)$, the second-order time-derivative of $\Psi(t)$ exists in $\Hil$. Separating out the electric components, $\partial_t^2 \Psi(t) + M^2 \Psi(t) = 0$ yields~\eqref{setup:eqn:wave_equation_electric:dynamics}. By definition of $\Hil$ and the transversality of the initial condition, also the second initial condition $\partial_t \psi^E(t_0) = - \mu^{-1} \, \nabla \times \psi^H(t_0)$ of the wave equation is satisfied. Hence, the magnetic and electric part satisfy \eqref{setup:eqn:wave_equation_electric:dynamics}. Lastly, the transversality condition~\eqref{setup:eqn:wave_equation_electric:constraint} is preserved (Proposition~\ref{setup:prop:complex_representation_real_fields}~(2)). This shows that such a strong solution to Maxwell's equations yields a strong solution to the wave equation~\eqref{setup:eqn:wave_equation_electric} for the electric field.

	Now conversely, suppose $\psi^E(t)$ solves \eqref{setup:eqn:wave_equation_electric}. Our additional assumption $\Phi \in \mathcal{D}(M^2)$ guarantees that it is indeed a strong solution as the first- and second-order time-derivatives exist in $L^2_{\eps}(\R^3,\C^3)$. Writing \eqref{setup:eqn:wave_equation_electric} as a first-order equation yields

	\begin{align*}
		\frac{\partial}{\partial t} \left (

		\begin{matrix}
			\psi^E(t) \\
			\eta^E(t) \\
		\end{matrix}

		\right ) = \left (

		\begin{matrix}
			0 & \id \\
			M_{EE}^2 & 0 \\
		\end{matrix}

		\right ) \left (

		\begin{matrix}
			\psi^E(t) \\
			\eta^E(t) \\
		\end{matrix}

		\right )
		.
	\end{align*}

	However, instead of using $\eta^E(t) = \partial_t \psi^E(t)$ as the second variable, we can introduce the magnetic field $\psi^H(t) := \bigl ( \nabla^{\times} \bigr )^{-1} \, \eps \, \eta^H(t)$. This change of variables makes sense as $\eta^E(t) \in \ker \, ( \nabla \cdot \eps )$ and the curl has the inverse $\bigl ( \nabla^{\times} \bigr )^{-1} : \ker (\nabla \cdot \,) \longrightarrow L^2_{\eps}(\R^3,\C^3)$ for divergence-free fields. Using the complex magnetic field as the second variable, we obtain the equations

	\begin{align*}
		\left (

		\begin{matrix}
			\id & 0 \\
			0 & \eps^{-1} \, \nabla^{\times} \\
		\end{matrix}

		\right ) \; \frac{\partial}{\partial t} \left (

		\begin{matrix}
			\psi^E(t) \\
			\psi^H(t) \\
		\end{matrix}

		\right )
		= \left (

		\begin{matrix}
			0 & \id \\
			M_{EE}^2 & 0 \\
		\end{matrix}

		\right )
		\left (

		\begin{matrix}
			\id & 0 \\
			0 & \eps^{-1} \, \nabla^{\times} \\
		\end{matrix}

		\right ) \left (

		\begin{matrix}
			\psi^E(t) \\
			\psi^H(t) \\
		\end{matrix}

		\right )
		,
	\end{align*}

	which can be rewritten as the dynamical Maxwell equation~\eqref{setup:eqn:Maxwell_equations:dynamics}. The latter step is once again allowed because $\bigl ( \nabla^{\times} \bigr )^{-1} \, \eps : \ker \, ( \nabla \cdot \eps ) \longrightarrow L^2_{\eps}(\R^3,\C^3)$ is bounded. Moreover, with our specific choice of initial condition $\Phi = Q (\mathbf{E}_0,\mathbf{H}_0) \in \Hil$, the resulting electromagnetic field $\Psi(t) = \bigl ( \psi^E(t) , \psi^H(t) \bigr )$ indeed is a positive frequency wave.

	This is the proof of the statement for strong solutions. But of course, the extra assumption $\Phi \in \domain(M^2)$ on the initial condition can be dropped if we work with weak solutions. In essence, we impose the extra condition on the test functions $\Theta \in \domain(M^2)$ and exploit that $\domain(M^2)$ lies densely in $\Hil$.
\end{proof}

\begin{remark}[Extension to other dimensions]\label{setup:rem:other_dimensions}
	Evidently, none of our arguments rely on the fact that the spatial domain is \emph{all} of $\R^3$ rather than some subset. We may want to work on a subdomain of the form $\R^2 \times [0,h] \subset \R^3$ to model a quasi-2d waveguide slab, for example. Here, a proper choice of boundary conditions such as those for a perfect electric or magnetic conductor on the upper and lower plate are necessary to define Maxwell's equations — and, by extension, the Maxwell operators.

	Consequently, the equivalence of Chern numbers applies to two- and three-dimensional topological photonic crystals alike.
\end{remark}