2022/06/11 by Steve Zelditch, Zelditch, Steve
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2206.05574
openalex publication_date 2022/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article is about two types of restrictions of eigenfunctions ϕj on a compact Riemannian manifold (M,g): First, we restrict to a submanifold H ⊂ M, and expand the restriction γH ϕj in eigenfunctions ek of H. We then Fourier restrict γH ϕj to a short interval of eigenvalues of H. Laplace eigenvalues of M are denoted λj2 and those of H are denoted μk2. The Fourier coefficients are negligible unless the H- eigenvalues lie in the interval μk ∈ [-λj, λj]. The short windows have the form |μk - c λj| < ε. The goal is to obtain asymptotics and estimates of the Fourier coefficients of γH ϕj and to see how they vary with c. In prior work with E. L. Wyman and Y. Xi, we obtained asymptotics for sums over (μk, λj) 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.