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Uniform distribution of eigenfunctions on compact hyperbolic surfaces. (English) Zbl 0643.58029
Let X be a compact Riemann surface with a metric of constant curvature - 1. Suppose that \(\phi_ k\) are the eigenfunctions of the Laplace operator of X. The author proves that almost all of these eigenfunctions become uniformly distributed over X, as \(k\to \infty\). In fact, after passing to a subsequence of density one, we have \(\int_{E}\phi\) \(2_ k\to vol(E)/vol(X)\). This result is developed in a more general framework, using a suitable calculus of pseudodifferential operators.
Reviewer: H.Donnelly

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
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