Quantum Ergodicity on large hyperbolic surfaces for local and pseudolocal operators

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.