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.


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)
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