Concentration of symmetric eigenfunctions. (English) Zbl 1193.58016
Summary: We examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an application of these results, we give a simple proof of the fact that in a manifold of constant positive sectional curvature, every measure which is invariant by the geodesic flow is an invariant semiclassical measure.
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
Keywords:
eigenfunctions of the Laplacian; semiclassical measures; Wigner distributions; manifolds of constant sectional curvature; invariant measuresReferences:
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