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This paper systematically develops the Schr\"odinger formalism that is validalso for gyrotropic media where the material weights $W = \left (\begin{smallmatrix} \varepsilon & \chi \chi^* & \mu \end{smallmatrix} \right )\neq \overline{W}$ are complex. This is a non-trivial extension of theSchr\"odinger formalism for non-gyrotropic media (where $W = \overline{W}$)that has been known since at least the 1960s. Here, Maxwell's equations arerewritten in the form $\mathrm{i} \partial_t \Psi = M \Psi$ where theselfadjoint (hermitian) Maxwell operator $M = W^{-1} \, \mathrm{Rot} \, \big|_{\omega \geq 0} = M^*$ takes the place of the Hamiltonian and $\Psi$ is acomplex wave representing the physical field $(\mathbf{E},\mathbf{H}) = 2\mathrm{Re} \, \Psi$. Writing Maxwell's equations in Schr\"odinger form givesus access to the rich toolbox of techniques initially developed for quantummechanics and allows us to apply them to classical waves. To show its utility,we explain how to identify conserved quantities in this formalism. Moreover, wesketch how to extend our ideas to other classical waves.

The Schr\"odinger Formalism of Electromagnetism and Other Classical Waves --- How to Make Quantum-Wave Analogies Rigorous