Quantum Mixing for Schr\"odinger eigenfunctions in Benjamini-Schramm limit

Abstract

Let $\{X_n\}_{n\in\mathbb{N}}$ be a sequence of compact hyperbolic surfaces which is uniformly discrete and Benjamini-Schramm converges to $\mathbb{H}$ and let $\{V_n\}_{n\in\mathbb{N}}$ be a sequence of potentials such that the $L^2$-norm of $V_n$ is $o(1)$ with respect to the volume of $X_n$. We prove quantum mixing for the eigenfunctions of $-\Delta_{X_n}+V_n$ in any sufficiently large spectral window $I$ for bounded observables which are polynomial mixing with respect to the geodesic flow. Examples of such sequences of potentials include point-cloud potentials just below the thermodynamic limit, Hartree potentials for dilute Bose gases in hyperbolic space, and sequences induced by bounded potentials in $L^p(\mathbb{H})$ for some $p>0$. This is the first result of this kind beyond the locally symmetric setting. The proof combines classical geodesic flow mixing on $T^1X_n$, replacing Nevo's ergodic theorem, with the Duhamel formula and recently developed geometric wave-kernel estimates by the first author.