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Geodesic biangles and Fourier coefficients of restrictions of eigenfunctions

Emmett L. Wyman, Yakun Xi and Steve Zelditch

Vol. 4 (2022), No. 4, 675–725
Abstract

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ϕj of a compact Riemannian manifold to a submanifold H M. We fix a number c (0,1) and study the asymptotics of the thin sums,

N𝜖,Hc(λ) := j,λjλ k:|μkcλj|<𝜖|Hϕjψ¯k dV H|2,

where {λj} are the eigenvalues of ΔM, and {(μk,ψk)} are the eigenvalues and the corresponding eigenfunctions of ΔH. The inner sums represent the “jumps” of N𝜖,Hc(λ) and reflect the geometry of geodesic c-biangles with one leg on H and a second leg on M with the same endpoints and compatible initial tangent vectors ξ SHcM, πHξ BH, where πHξ is the orthogonal projection of ξ to H. A c-biangle occurs when |πHξ||ξ| = c. Smoothed sums in μk are also studied and give sharp estimates on the jumps. The jumps themselves may jump as 𝜖 varies, at certain values of 𝜖 related to periodicities in the c-biangle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of our previous article, where the inner sums run over k such that |μkλj c| 𝜖 and where geodesic biangles do not play a role.

Keywords
restriction of eigenfunctions, fuzzy ladder projectors, Kuznecov formula
Mathematical Subject Classification
Primary: 35S30, 58J40
Milestones
Received: 26 May 2021
Revised: 8 December 2021
Accepted: 22 March 2022
Published: 21 January 2023
Authors
Emmett L. Wyman
Department of Mathematics
Northwestern University
Evanston, IL
United States
Yakun Xi
School of Mathematical Sciences
Zhejiang University
Hangzhou
China
Steve Zelditch
Department of Mathematics
Northwestern University
Evanston, IL
United States