Restriction of eigenfunctions to totally geodesic submanifolds

This article is about two types of restrictions of eigenfunctions $\phi_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $\gamma_H \phi_j$ in eigenfunctions $e_k$ of $H$. We then Fourier restrict $\gamma_H \phi_j$ to a short interval of eigenvalues of $H$. Laplace eigenvalues of $M$ are denoted $\lambda_j^2$ and those of $H$ are denoted $\mu_k^2$. The Fourier coefficients are negligible unless the $H$- eigenvalues lie in the interval $\mu_k \in [-\lambda_j, \lambda_j]$. The short windows have the form $|\mu_k - c \lambda_j| < \epsilon$. The goal is to obtain asymptotics and estimates of the Fourier coefficients of $\gamma_H \phi_j$ and to see how they vary with $c$. In prior work with E. L. Wyman and Y. Xi, we obtained asymptotics for sums over $(\mu_k, \lambda_j)$ in such windows for $0 < c < 1$. In this article, we obtain `edge' asymptotics when $c=1$ and $H$ is totally geodesic. The order of magnitude and leading coefficient are very different from the case $c<1$. In particular, they depend on the dimension of $H$. We explain how to bridge the bulk results and edge results.