On the distribution of perturbations of propagated Schrödinger eigenfunctions (Q402282)

Summary: Let \((M,g_0)\) be a compact Riemmanian manifold of dimension \(n\). Let \(P_0 (h) := -h^2\Delta_{g}+V\) be the semiclassical Schrödinger operator for \(h \in (0,h_0]\), and let \(E\) be a regular value of its principal symbol. Write \(\phi_h\) for an \(L^2\)-normalized eigenfunction of \(P_0(h)\) with eigenvalue \(E(h) \in [E-o(1),E+ o(1)]\). We consider a smooth family of metric perturbations \(g_u\) of \(g_0\) with \(u\) in the product space \(B^k(\varepsilon)=(-\varepsilon,\varepsilon)^k\subseteq \mathbb{R}^k\) satisfying a certain admissibility condtition. For \(P_{u}(h) := -h^2 \Delta_{g_u} +V\) and small \(|t|>0\), we define the propagated perturbed eigenfunctions \[ \varphi_{h,t}^{(u)}:=e^{-\frac{i}{h}t P_u(h)} \varphi_h. \] 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 \[ \text{Re}(\varphi^{(\cdot)}_{h,t}(x)): B^{k}(\varepsilon) \to \mathbb{R},\;\;x\in M. \] In particular, under an admissibility condition on the metric when \((M,g)\) is chaotic, we compute the \(h \to 0^+\) asymptotics of the variance \(\text{Var}[\text{Re}(\varphi^{(\cdot)}_{h,t}(x))] \) and show that the odd moments vanish as \(h \to 0^+\) as long as \(x\) is not on the generalized cuastic set where \(V(x)=E\).