Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

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 $c\in (0,1)$ and study the asymptotics of the thin sums, $$N_{\epsilon ,H}^c(\lambda ):=\sum _{j,{\lambda }_j\le \lambda }\sum _{k:|{\mu }_k-c{\lambda }_j|<\epsilon }{\left|{\int }_H{\varphi }_j\overline{{\psi }_k}d{V}_H\right|}^2$$ where $\{{\lambda }_j\}$ are the eigenvalues of ${\sqrt{-\Delta }}_M,$ and $\{({\mu }_k,{\psi }_k)\}$ 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 $\xi \in S_H^{c}M,{\pi }_H\xi \in B^{\ast }H$, where ${\pi }_H\xi$ is the orthogonal projection of $\xi$ to $H$. A c-bi-angle occurs when $\frac{|{\pi }_H\xi |}{|\xi |}=c$. Smoothed sums in ${\mu }_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $\epsilon$ 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 $k:|\frac{{\mu }_k}{{\lambda }_j}-c|\le \epsilon$ and where geodesic bi-angles do not play a role.

51 pages. Referee's comments incorporated. To appear in Pure and Applied Analysis

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