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

TL;DR

This paper investigates how small perturbations of the metric on a compact Riemannian manifold affect the distribution of propagated Schr"odinger eigenfunctions, revealing asymptotic variance behavior and moment vanishing in the semiclassical limit.

Contribution

It provides a detailed analysis of the distribution of perturbed eigenfunctions under metric variations, including variance asymptotics and moment properties in the semiclassical regime.

Findings

Asymptotic behavior of the variance of real parts of perturbed eigenfunctions.

Vanishing of all odd moments as the semiclassical parameter tends to zero.

Results are specific to ergodic manifolds, linking dynamics to eigenfunction distribution.

Abstract

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 $h \to 0^+.$