Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

DOI10.2140/PAA.2022.4.675arXiv2104.09470MaRDI QIDQ6365653

E. Wyman, Yakun Xi, Steve Zelditch

Publication date: 19 April 2021

Abstract: This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions $ph{i}_j$ of a compact Riemannian manifold to a submanifold $HsubsetM$. We fix a number $cin(0,1)$ and study the asymptotics of the thin sums, ${\displaystyle }$ N^{c} _{epsilon, H }(lambda): = sum_{j, lambda_j leq lambda} sum_{k: |mu_k - c lambda_j | < epsilon} left| int_{H} phi_j overline{psi_k}dV_H ight|^2 ${\displaystyle }$ where $lambd{a}_j$ are the eigenvalues of $sqrt{-Delta}_M,$ and $(m{u}_k,ps{i}_k)$ are the eigenvalues, resp. eigenfunctions, of $sqrt{-Delta}_H$. The inner sums represent the `jumps' of $N_{epsilon,H}^c(lambda)$ and reflect the geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $xiin{S}_H^{c}M,p{i}_{H}xiin{B}^{*}H$, where $p{i}_{H}xi$ is the orthogonal projection of $xi$ to $H$. A c-bi-angle occurs when $frac|p{i}_{H}xi||xi|=c$. Smoothed sums in $m{u}_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $epsilon$ varies, at certain values of $epsilon$ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over $k:|fracm{u}_{k}lambd{a}_j-c|leqepsilon$ and where geodesic bi-angles do not play a role.

Mathematics Subject Classification ID

Pseudodifferential and Fourier integral operators on manifolds (58J40) Fourier integral operators applied to PDEs (35S30)

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