\begin{equation}\label{eq:forward_eigen}
\begin{split}
    \frac{\partial}{\partial t}e^{-\lambda t} \mathcal{F}_n(x)&={\cal L}e^{-\lambda t} \mathcal{F}_n(x)\\
    -\lambda \mathcal{F}_n(x)&= \lambda_n \mathcal{F}_n(x)\,.
\end{split}
\end{equation}
