On the distribution of perturbations of propagated Schr\"odinger eigenfunctions

Let $(M,g_0)$ be a compact Riemmanian manifold of dimension $n$. Let $P_0 (\h) := -\h^2\Delta_{g}+V$ be the semiclassical Schr\"{o}dinger operator for $\h \in (0,\h_0]$, and let $E$ be a regular value of its principal symbol $p_0(x,\xi)=|\xi|^2_{g_0(x)} +V(x)$. Write $\varphi_\h$ for an $L^2$-normalized eigenfunction of $P(\h)$, $P_0(\h)\varphi_\h =E(\h)\varphi_\h$ and $E(\h) \in [E-o(1),E+ o(1)]$. Consider a smooth family of perturbations $g_u$ of $g_0$ with $u$ in the ball $\mathcal B^k(\varepsilon) \subset \mathbb R^k$ of radius $\varepsilon>0$. For $P_{u}(\h) := -\h^2 \Delta_{g_u} +V$ and small $|t|$, we define the propagated perturbed eigenfunctions $$\varphi_\h^{(u)}:=e^{-\frac{i}{\h}t P_u(\h)} \varphi_\h.$$ We study the distribution of the real part of the perturbed eigenfunctions regarded as random variables $$\Re (\varphi^{(\cdot)}_\h(x)):\mathcal B^{k}(\varepsilon) \to \mathbb R \quad \quad \text{for}\;\, x\in M.$$ In particular, when $(M,g)$ is ergodic, we compute the $h \to 0^+$ asymptotics of the variance $\text{Var} [\Re (\varphi^{(\cdot)}_\h(x))]$ and show that all odd moments vanish as $h \to 0^+.$