×

Quantum ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces. (English) Zbl 1384.37035

Let \(X=\Gamma \backslash \mathbb H\) be a connected compact hyperbolic surface, \(\lambda_0\leq \lambda_1\leq \cdots\) be the sequence of eigenvalues of the Laplacian on \(X\), \(\{\psi_k\,,k\in\mathbb N\}\) be the corresponding orthonormal basis of eigenfunctions in \(L^2\), and \(\,dx\,\), \(\,r_X(x)\) and \(\ell(X):= \min r_X\) respectively denote the normalized volume measure on \(X\), the injectivity radius of \(X\) at any \(x\in X\) and the injectivity radius of \(X\). Let \(I\) be a compact sub-interval of \(\,]1/4,\infty[\,\). Then the authors establish the following main result, which gives a quantitative account of the equidistribution of the eigenfunctions \(\psi_j\,\): there exists \(\,C(I)>0\) and \(\varrho(\lambda_1)>0\,\) such that for any \(\,\varphi\in L^\infty(X)\) and any \(R>C(I)\) we have \[ C(I)^{-1}\sum_{\lambda_j\in I} \bigg| \big\langle \psi_j,\varphi\,\psi_j\big\rangle -\int_X \varphi(x) dx \bigg|^2 \leq\, {\|\varphi\|_2^2\over \varrho(\lambda_1)^2R} + {e^{4R}\over \ell(X) }\mathrm{Vol}\big(\{x\in X\,|\, r_X(x)<R\}\big) \|\varphi\|_\infty^2\, . \] Then the authors apply this to a sequence \((X_n)\) of compact hyperbolic surfaces, which are supposed to possess uniform injectivity radius \(\,\ell:= \inf_n\limits \ell(X_n)>0\,\) and spectral gap \(\,\beta:= \inf_n\limits\lambda_1(X_n)>0\,\), and to satisfy the following Benjamini-Schramm convergence: \[ \lim_{n\to\infty}\limits {\mathrm{Vol}(\{x\in X_n\,|\, r_{X_n}(x)<R\})\over \mathrm{Vol}(X_n)} = 0\, \] for any \(R>0\). Thus they obtain \[ \sum_{\lambda_j(X_n)\in I} \bigg| \big\langle \psi_j^{(n)},\varphi_n\,\psi_j^{(n)}\big\rangle -\int_{X_n}\! \varphi_n(x) dx \bigg|^2 = o\big(N(X_n,I)\big)\quad \text{ as }n\to \infty, \] for any uniformly bounded sequence \(\,\varphi_n\in L^\infty(X_n)\), where \( N(X_n,I)\) counts the eigenvalues of \(X_n\) in \(I\). Their proof is inspired by recent works on discrete graphs and does not use any microlocal analysis, which allows the authors to consider observables with minimal regularity.

MSC:

37D05 Dynamical systems with hyperbolic orbits and sets
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
81Q50 Quantum chaos
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid