Quantum ergodic restriction theorems, II: manifolds without boundary

Abstract

We prove that if $(M,g)$ is a compact Riemannian manifold with ergodic geodesic flow, and if $H\subset M$ is a smooth hypersurface satisfying a generic asymmetry condition with respect to the geodesic flow, then restrictions ${\varphi }_j{|}_H$ of an orthonormal basis $\{{\varphi }_j\}$ of $\Delta$-eigenfunctions of $(M,g)$ to $H$ are quantum ergodic on $H$. The condition on $H$ is satisfied by geodesic circles, closed horocycles and generic closed geodesics on a hyperbolic surface.

53 pages. Second in a series. The paper is self-contained; the methods and results are independent of the first article (arXiv:1005.1636), which dealt with Euclidean domains with ergodic billiards. Some clarifications and an appendix added extending the result to the semi-classical case

References in corpus (1)

• Concerning the $L^4$ norms of typical eigenfunctions on compact surfaces

Cited by in corpus (6)

• The quantum unique ergodicity conjecture for thin sets
• Eigenfunctions and Nodal Sets
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• $C^{\infty }$ spectral rigidity of the ellipse
• Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces
• Matrix elements of Fourier Integral Operators