Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces
| Autor: | Tuomas Sahlsten, Etienne Le Masson |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
General Mathematics Hyperbolic geometry math-ph Microlocal analysis FOS: Physical sciences Dynamical Systems (math.DS) 81Q50 37D40 11F72 01 natural sciences Benjamini–Schramm convergence Mathematics - Spectral Theory Selberg transform math.MP 0103 physical sciences FOS: Mathematics quantum ergodicity Ergodic theory Mathematics - Dynamical Systems 0101 mathematics Spectral Theory (math.SP) Mathematical Physics Eigenvalues and eigenvectors Mathematics eigenfunctions of the Laplacian 37D40 010102 general mathematics math.SP Propagator Mathematical Physics (math-ph) Mathematics::Spectral Theory Eigenfunction Quantum chaos 11F72 81Q50 hyperbolic dynamics rate of mixing short geodesics 010307 mathematical physics Quantum ergodicity mean ergodic theorem math.DS quantum chaos |
| Zdroj: | Duke Math. J. 166, no. 18 (2017), 3425-3460 Duke Mathematical Journal Le Masson, E & Sahlsten, T 2017, ' Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces ', Duke Mathematical Journal, vol. 166, no. 18, pp. 3425-3460 . https://doi.org/10.1215/00127094-2017-0027 Sahlsten, T & Le Masson, E 2017, ' Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces ', Duke Mathematical Journal, vol. 166, no. 18, pp. 3425-3460 . https://doi.org/10.1215/00127094-2017-0027 |
| ISSN: | 0012-7094 |
| DOI: | 10.1215/00127094-2017-0027 |
| Popis: | 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. Comment: v3: 25 pages, 3 figures, the proof in Section 9 has been corrected, to appear in Duke Math. J |
| Databáze: | OpenAIRE |
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