Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces

TL;DR

This paper proves a quantum ergodicity theorem for hyperbolic surfaces converging to the hyperbolic plane, using wave propagation and averaging operators, addressing questions in spectral geometry and Maass forms.

Contribution

It introduces a wave propagation approach for quantum ergodicity on hyperbolic surfaces that avoids microlocal analysis and applies to sequences converging in the Benjamini-Schramm sense.

Findings

Quantum ergodicity holds for fixed spectral windows on converging hyperbolic surfaces.

The method simplifies analysis by replacing wave propagators with averaging operators.

Results apply to eigenfunctions on arithmetic surfaces and Maass forms.

Abstract

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdi\`{e}re. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.