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Ergodicity and eigenfunctions of the Laplacian. (Ergodicité et fonctions propres du laplacien.) (French) Zbl 0592.58050
The author proves a statement of A. Shnirel’man: Let $$M$$ be a compact Riemannian manifold with ergodic geodesic flow. Let $$(\phi_ k)$$ denote an orthonormal basis of eigenfunctions of the Laplace operator corresponding to the eigenvalues $$(\lambda_ k)$$. Let $$A$$ be a pseudo-differential operator of order 0 with principal symbol $$a$$. Then there exists a subsequence $$(\lambda_{k_ i})$$ such that $\lim <A\phi_{k_ i}, \phi_{k_ i}>=\int a\, d\omega,$ where $$<\,, \,>$$ denotes the $$L^ 2$$-inner-product and $$d\omega$$ the Liouville measure on the fiber of cotangent unit vectors.
Reviewer: Udo Simon (Berlin)

MSC:
 58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity 35P20 Asymptotic distributions of eigenvalues in context of PDEs 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 53C20 Global Riemannian geometry, including pinching
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References:
 [1] Colin de Verdiere, Y.: Invent. math.43, 15-52 (1977) · Zbl 0449.53040 [2] Colin de Verdiere, Y.: Duke Math. J.46, 169-182 (1978) · Zbl 0411.35073 [3] Schnirelman, A.: Usp. Mat. Nauk29, 181-182 (1974) [4] Taylor, M.: Pseudo-differential operators. Princeton, NJ: Princeton University Press 1981 · Zbl 0482.34021 [5] Zelditch, S.: Eigenfunctions on compact Riemann-surfaces ofg?2. Preprint 1984 (New York)
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