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

TL;DR

This paper proves quantum mixing of eigenfunctions for Schr"odinger operators on hyperbolic surfaces converging to the hyperbolic plane, with applications to various geometric and physical models.

Contribution

It establishes quantum mixing results for eigenfunctions on hyperbolic surfaces in the Benjamini-Schramm limit, extending to random and arithmetic covers.

Findings

Quantum mixing holds for eigenfunctions in large spectral windows.

Results apply to large degree lifts and random hyperbolic surfaces.

Method combines Duhamel formula with exponential mixing of geodesic flow.

Abstract

Let $-\Delta_{\mathbb{H}}+V$ be the Schr\"odinger 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 $-\Delta_{X_n}+V_n$ 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 $T^1 X_n$.