Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

John A. Toth; Steve Zelditch

doi:10.1007/s00039-013-0220-0

Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

John A. Toth, Steve Zelditch^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

35 Scopus citations

Abstract

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 φ_j{pipe}_H of an orthonormal basis {φ_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 〈 Fφ_j, φ_j〉 of Fourier integral operators F whose canonical relation almost nowhere commutes with the geodesic flow must tend to zero.

Original language	English (US)
Pages (from-to)	715-775
Number of pages	61
Journal	Geometric and Functional Analysis
Volume	23
Issue number	2
DOIs	https://doi.org/10.1007/s00039-013-0220-0
State	Published - Apr 2013

Funding

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.

ASJC Scopus subject areas

Analysis
Geometry and Topology

Access to Document

10.1007/s00039-013-0220-0

Cite this

@article{a5b3fa78418f431f8addcaa2a2e08c75,

title = "Quantum Ergodic Restriction Theorems: Manifolds Without Boundary",

abstract = "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.",

author = "Toth, {John A.} and Steve Zelditch",

note = "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.",

year = "2013",

month = apr,

doi = "10.1007/s00039-013-0220-0",

language = "English (US)",

volume = "23",

pages = "715--775",

journal = "Geometric and Functional Analysis",

issn = "1016-443X",

publisher = "Birkhauser Verlag Basel",

number = "2",

}

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.

UR - http://www.scopus.com/inward/record.url?scp=84877586326&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84877586326&partnerID=8YFLogxK

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 -

Quantum Ergodic Restriction Theorems: Manifolds Without Boundary

Abstract

Funding

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this