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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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