Pure Appl. Anal. 4 (4), 675-725, (2022) DOI: 10.2140/paa.2022.4.675
Emmett L. Wyman, Yakun Xi, Steve Zelditch
KEYWORDS: restriction of eigenfunctions, fuzzy ladder projectors, Kuznecov formula, 35S30, 58J40
This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions of a compact Riemannian manifold to a submanifold . We fix a number and study the asymptotics of the thin sums,
where are the eigenvalues of , and are the eigenvalues and the corresponding eigenfunctions of . The inner sums represent the “jumps” of and reflect the geometry of geodesic -biangles with one leg on and a second leg on with the same endpoints and compatible initial tangent vectors , , where is the orthogonal projection of to . A -biangle occurs when . Smoothed sums in are also studied and give sharp estimates on the jumps. The jumps themselves may jump as varies, at certain values of related to periodicities in the -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 such that and where geodesic biangles do not play a role.