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

Let $(M,g_0)$ be a compact Riemmanian manifold of dimension $n$. Let $P_0 (\h) := -\h^2\Delta_{g}+V$bethesemiclassicalSchr\ddot{o}dingeroperatorfor$\h \in (0,\h_0]$,andlet$E$bearegularvalueofitsprincipalsymbol$p_0(x,\xi)=|\xi|^2_{g_0(x)} +V(x)$\mathrm{.}Write$\varphi_\h$foran$L^2$-normalized eigenfunction of $P(\text{h})$, $P_0(\text{h}){\phi }_{\text{h}}=E(\text{h}){\phi }_{\text{h}}$ and $E(\h) \in [E-o(1),E+ o(1)]$\mathrm{.}Considerasmoothfamilyofperturbations$g_u$of$g_0$with$u$intheball$\mathcal B^k(\varepsilon) \subset \mathbb R^k$ofradius$\varepsilon>0$\mathrm{.}For$P_{u}(\h) := -\h^2 \Delta_{g_u} +V$andsmall$|t|$, we define the propagated perturbed eigenfunctions $${\phi }_{\text{h}}^{(u)}:=e^{-\frac{i}{\text{h}}t{P}_u(\text{h})}{\phi }_{\text{h}}\mathrm{.}$$ We study the distribution of the real part of the perturbed eigenfunctions regarded as random variables $$\mathrm{\Re }({\phi }_{\text{h}}^{(\cdot )}(x)):{\mathcal{B}}^k(\epsilon )\to \mathbb{R}\text{for}x\in M\mathrm{.}$$ In particular, when $(M,g)$ is ergodic, we compute the $h\to 0^{+}$ asymptotics of the variance $\text{Var}[\mathrm{\Re }({\phi }_{\text{h}}^{(\cdot )}(x))]$ and show that all odd moments vanish as $h\to 0^{+}\mathrm{.}$