this notation, it is easy to show that Eqs.(\[eqn:discrTMx\]) and (\[eqn:discrTMy\]) reduce to $${\frac{\partial}{\partial t}}\Psi(t)=H \Psi(t),$$ where the matrix $H$ is given by $$H=
\mathop{{\sum}'}_{i=1}^{n} \left[
\beta_{i+1,i}
\left(
\be^{{\phantom{T}}}_{i}\be^T_{i+1}-
\be^{{\phantom{T}}}_{i+1}\be^T_{i}\right)+
\beta_{i+1,i+2}
\left(
\be^{{\phantom{T}}}_{i+1}\be^T_{i+2}-
\be^{{\phantom{T}}}_{i+2}\be^T_{i+1}\right)\right]\,,
\label{eqn:oper_1D}$$ with $\beta_{i,j}=1/(\delta\sqrt{\varepsilon_i \mu_j})$ and the prime indicates that the sum is over odd integers only. In complete analogy to Eq. (\[eqn:formal\]) the time evolution of $\Psi(t)$ is formally given by $\Psi(t) = U(t) \Psi(0)$ with $U(t) =\exp(t{H})$.

The notation introduced above will prove most useful for the case of 2D and 3D for which it is rather cumbersome to write down matrix representations. For the 1D case it is not difficult and in fact very instructive to write down the matrix $H$ explicitly. Indeed, we have $$H=
\left( \begin{array}{ccccc}
0 & \beta_{2,1} & & & \\
-\beta_{2,1} & 0 & \beta_{2,3} & & \\
& \ddots & \ddots & \ddots & \\
& & -\beta_{n-1,n-2} & 0 & \beta_{n-1,n} \\
& & & -\beta_{n-1,n} & 0
\end{array} \right),
\label{H}$$ and we immediately see that $H$ is skew-symmetric by construction. Furthermore, for $n$ odd we have $${\frac{\partial}{\partial t}}\Psi(1,t)=\beta_{2,1}
\Psi(2,t)\;\;\;\;\;{\rm and}\;\;\;\;\;
{\frac{\partial}{\partial t}}\Psi(n,t)=-\beta_{n-1,n}
\Psi(n-1,t)\,,$$ such that the electric field vanishes at the boundaries ($Y_z(0,t)=Y_z(n+1,t)=0$, see also Fig. 