  Consider the linear transport equation in $1$D under an external confining
potential $\Phi$: \begin{equation*} \partial_t f + v \partial_x f - \partial_x
\Phi \partial_v f = 0. \end{equation*} For $\Phi = \frac{x^2}{2} +
\frac{\epsilon x^4}{2}$ (with $\epsilon >0$ small), we prove phase mixing and
quantitative decay estimates for $\partial_t \varphi := - \Delta^{-1}
\int_{\mathbb{R}} \partial_t f \, \mathrm{d} v$, with an inverse polynomial
decay rate $O(\langle t\rangle^{-2})$. In the proof, we develop a commuting
vector field approach, suitably adapted to this setting. We will explain why we
hope this is relevant for the nonlinear stability of the zero solution for the
Vlasov--Poisson system in $1$D under the external potential $\Phi$.
