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)
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