Quantum Mixing for Schrödinger eigenfunctions in Benjamini-Schramm limit

Abstract

Let $-\hat{I}{”}_{\mathbb{H}}+V$ be the Schrödinger operator on $\mathbb{H}$ where $V\in L^p(\mathbb{H})\cap L^{\infty }(\mathbb{H})$ for some $p>0$. If $(X_n)$ is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to $\mathbb{H}$, we prove quantum mixing for the eigenfunctions of $-\hat{I}{”}_{X_n}+V_n$ in any sufficiently large spectral window $I$, where $V_n$ is the potential on $X_n$ induced by $V$. These apply to large degree lifts of a potential on a base surface such as congruence covers of arithmetic surfaces, with high probability to random hyperbolic surfaces in the Weil-Petersson model of large genus, and to Hartree one-particle operators arising in thermodynamic limit of many-body Bose gas on hyperbolic surfaces. The proof uses the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on $T^1{X}_n$.

36 pages