Recent Issues Volume 4, Issue 4 Volume 4, Issue 3 Volume 4, Issue 2 Volume 4, Issue 1 Volume 3, Issue 4 Volume 3, Issue 3 Volume 3, Issue 2 Volume 3, Issue 1 Volume 2, Issue 4 Volume 2, Issue 3 Volume 2, Issue 2 Volume 2, Issue 1 Volume 1, Issue 4 Volume 1, Issue 3 Volume 1, Issue 2 Volume 1, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN (electronic): 2578-5885 ISSN (print): 2578-5893 Author Index To Appear Other MSP Journals
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 ${\varphi }_{j}$ of a compact Riemannian manifold to a submanifold $H\subset M$. We fix a number $c\in \left(0,1\right)$ and study the asymptotics of the thin sums,

 ${N}_{𝜖,H}^{c}\left(\lambda \right):=\sum _{j,{\lambda }_{j}\le \lambda }\sum _{k:|{\mu }_{k}-c{\lambda }_{j}|<𝜖}|{\int }_{H}{\varphi }_{j}{\overline{\psi }}_{k}d{V}_{H}{|}^{2},$

where $\left\{{\lambda }_{j}\right\}$ are the eigenvalues of ${\sqrt{-\mathrm{\Delta }}}_{M}$, and $\left\{\left({\mu }_{k},{\psi }_{k}\right)\right\}$ are the eigenvalues and the corresponding eigenfunctions of ${\sqrt{-\mathrm{\Delta }}}_{H}$. The inner sums represent the “jumps” of ${N}_{𝜖,H}^{c}\left(\lambda \right)$ 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 $\xi \in {S}_{H}^{c}M$, ${\pi }_{H}\xi \in {B}^{\ast }H$, where ${\pi }_{H}\xi$ is the orthogonal projection of $\xi$ to $H$. A $c$-biangle occurs when $|{\pi }_{H}\xi |∕|\xi |=c$. Smoothed sums in ${\mu }_{k}$ 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 $|{\mu }_{k}∕{\lambda }_{j}-c|\le 𝜖$ and where geodesic biangles do not play a role.

Keywords
restriction of eigenfunctions, fuzzy ladder projectors, Kuznecov formula
Mathematical Subject Classification
Primary: 35S30, 58J40