Quantum Ergodicity on large hyperbolic surfaces for local and pseudolocal operators

TL;DR

This paper proves a quantum ergodicity theorem for sequences of hyperbolic surfaces, extending previous results to local and pseudolocal operators under certain geometric and spectral conditions.

Contribution

It generalizes quantum ergodicity results from scalar observables to local and pseudolocal operators on hyperbolic surfaces with uniform geometric bounds.

Findings

Vanishing quantum variance on fixed spectral windows

Extension of ergodicity to pseudolocal operators

Applicable to sequences converging to the Poincaré disc

Abstract

We prove a quantum ergodicity theorem for sequences of closed hyperbolic surfaces converging to the Poincar\'e disc in the Benjamini-Schramm sense. Assuming a uniform lower bound on the injectivity radius and a spectral gap, we establish vanishing of quantum variance on fixed spectral windows for a class of observables that contains differential operators and finite-propagation smooth operators. This generalises a result of Le Masson and Sahlsten from scalar observables to both local and 'pseudolocal' operator settings.