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)


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
Full Text: DOI


[1] Colin de Verdiere, Y.: Invent. math.43, 15-52 (1977) · Zbl 0449.53040 · doi:10.1007/BF01390202
[2] Colin de Verdiere, Y.: Duke Math. J.46, 169-182 (1978) · Zbl 0411.35073 · doi:10.1215/S0012-7094-79-04608-8
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.