Fourier coefficients of restrictions of eigenfunctions

Abstract

Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H\subset M$ be a submanifold and let $\{{\psi }_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions ${\gamma }_H{e}_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{({\mu }_k,{\lambda }_j){\}}_{j,k-0}^{\infty }$ of the (square roots of the) Laplacian ${\Delta }_M$ on $M$ and the Laplacian ${\Delta }_H$ on $H$ in a family of suitably `thick' regions in ${\mathbb{R}}^2$. Thick regions include (1) the truncated cone ${\mu }_k/{\lambda }_j\in [a,b]\subset (0,1)$ and ${\lambda }_j\le \lambda$, and (2) the slowly thickening strip $|{\mu }_k-c{\lambda }_j|\le w(\lambda )$ and ${\lambda }_j\le \lambda$, where $w(\lambda )$ is monotonic and $1\ll w(\lambda )\lesssim {\lambda }^{1-1/n}$. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.

arXiv comment36 pages. Referees' suggestions incorporated. To appear in Sci. China Math