Quantum ergodicity and BenjaminiSchramm convergence of hyperbolic surfaces
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 Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the firstnamed 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 firstnamed 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.05720
 Bibcode:
 2016arXiv160505720L
 Keywords:

 Mathematics  Spectral Theory;
 Mathematical Physics;
 Mathematics  Dynamical Systems;
 81Q50;
 37D40;
 11F72
 EPrint:
 v3: 25 pages, 3 figures, the proof in Section 9 has been corrected, to appear in Duke Math. J