Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

By J. Toth, Steve Zelditch. 2011

66 citations recorded by the source corpus.

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 microlocal asymmetry condition, then restrictions $${\varphi_j |_H}$$ of an orthonormal basis $${\{\varphi_j\}}$$ of Δ-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. A key step in the proof is that matrix elements $${\langle{F}\varphi_j, \varphi_{j}\rangle}$$ of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/23f5663c28be753c2980c3f442202049c07aaef0

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Number of nodal domains and singular points of eigenfunctions of negatively curved surfaces with an isometric involution (2013, methodology): “ Dirichlet data assert the quantum ergodicity of restrictions ϕj|H of eigenfunctions or their normal derivatives to hypersurfaces H ⊂ M. In this section we review the QER theorem for hypersurfaces of [TZ]. It is used in the proof of Theorem 1.1. As mentioned above, it does not apply to the restricti”