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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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