TY - JOUR
T1 - Quantum Ergodic Restriction Theorems
T2 - Manifolds Without Boundary
AU - Toth, John A.
AU - Zelditch, Steve
N1 - Funding Information:
J. A. Toth: Research partially supported by NSERC Grant # OGP0170280 and a William Dawson Fellowship. S. Zelditch: Research partially supported by NSF Grant # DMS-0904252.
PY - 2013/4
Y1 - 2013/4
N2 - We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and if H ⊂ M is a smooth hypersurface satisfying a generic microlocal asymmetry condition, then restrictions φj{pipe}H of an orthonormal basis {φj} 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 〈 Fφj, φj〉 of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.
AB - We prove that if (M, g) is a compact Riemannian manifold with ergodic geodesic flow, and if H ⊂ M is a smooth hypersurface satisfying a generic microlocal asymmetry condition, then restrictions φj{pipe}H of an orthonormal basis {φj} 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 〈 Fφj, φj〉 of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.
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U2 - 10.1007/s00039-013-0220-0
DO - 10.1007/s00039-013-0220-0
M3 - Article
AN - SCOPUS:84877586326
SN - 1016-443X
VL - 23
SP - 715
EP - 775
JO - Geometric and Functional Analysis
JF - Geometric and Functional Analysis
IS - 2
ER -