On the distribution of perturbations of propagated Schrödinger eigenfunctions

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. % $p_0(x,\xi )=|\xi {|}_{g_0(x)}^2+V(x)$. Write \( \varphi_\h \) for an $L^2$-normalized eigenfunction of \( P_0(\h) \) with eigenvalue \( E(\h) \in [E-o(1),E+ o(1)] \). Consider a smooth family of metric perturbations $g_u$ of $g_0$ with $u$ in the $k$-ball $B^k(\epsilon )\subset {\mathbb{R}}^k$ of radius $\epsilon >0$. For \( P_{u}(\h) := -\h^2 \Delta_{g_u} +V \) and small $|t|>0$, we define the propagated perturbed eigenfunctions

\[ \varphi_\h^{(u)}:=e^{-\frac{i}{\h}t P_u(\h) } \varphi_\h. \]

They appear in the mathematical description of the Loschmidt echo effect in physics. Motivated by random wave conjectures in quantum chaos, we study the distribution of the real part of the perturbed eigenfunctions regarded as random variables \( \Re (\varphi^{(\cdot)}_\h(x)): B^{k}(\varepsilon) \to \mathbb R \) for $x\in M$. In particular, when $(M,g)$ is chaotic, we compute the $h\to 0^{+}$ asymptotics of the variance \( \text{Var} [\Re(\varphi^{(\cdot)}_\h(x))] \) and show that the odd moments vanish as $h\to 0^{+}.$