Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

Abstract

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ϕ j of a compact Riemannian manifold to a submanifold H ⊂ M . We fix a number c ∈ (0, 1) and study the asymptotics of the thin sums,λ) := j,λj ≤λ k:|µ k −cλj |<ǫ H ϕ j ψ k dV H 2 where {λ j } are the eigenvalues of √ −∆ M , and {(µ k , ψ k )} are the eigenvalues, resp. eigenfunctions, of √ −∆ H . The inner sums represent the 'jumps' of N c ǫ,H (λ) 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 ξ ∈ S c H M, π H ξ ∈ B * H, where π H ξ is the orthogonal projection of ξ to H. A c-bi-angle occurs when |πH ξ| |ξ| = c. Smoothed sums in µ k are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as ǫ varies, at certain values of ǫ 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 [WXZ20] where the inner sums run over k : | µ k λj − c| ≤ ǫ and where geodesic bi-angles do not play a role.

On the growth of generalized Fourier coefficients of restricted eigenfunctions

“…We note that the theorem holds in more generality than this, as the collection {ψ h } does not necessarily consist of eigenfunctions. To the best of our knowledge, the only existing results in this direction are due to Wyman, Xi, and Zelditch [WXZ20,WXZ21], where the authors obtain asymptotics for sums of the norm-squares of the generalized Fourier coefficients over the joint spectrum. If we take our collection {ψ h } independent of h, we recover the weighted averages result in [CG19b, Theorem 6], which we demonstrate in Example 1.8.…”

Restriction of eigenfunctions to totally geodesic submanifolds

“…But in general the Fourier coefficients may vary erratically as λ j varies or as µ k varies, and to obtain asymptotics of Fourier coefficients it is usually necessary to study averages of squares of Fourier coefficients in both the µ k and λ j spectral parameters. We therefore study thin window or "ladder Kuznecov sums" in the sense of [WXZ21],…”

“…where the test function ψ ∈ S(R) (Schwartz class). It is shown in [WXZ21] that the sums decay rapidly if c > 1. The asymptotics for c = 0 were first determined in [Zel92] and those are those for 0 ≤ c < 1 are determined in [WXZ21] for any submanifold.…”

“…It is shown in [WXZ21] that the sums decay rapidly if c > 1. The asymptotics for c = 0 were first determined in [Zel92] and those are those for 0 ≤ c < 1 are determined in [WXZ21] for any submanifold. In this article, and its sequel [Z22], the asymptotics are studied for the 'edge case' c = 1 in the case where H is a totally geodesic submanifold.…”

“…The Kuznecov sums (1.2) involve two types of localization: (i) localization of ϕ j along H; and (ii) Fourier localization of ϕ j | H to a thin window of Fourier modes on H of frequencies µ j close to λ j . In [WXZ21] the first sum in (1.2) is called 'smooth-sharp' since it involves the smoothed inner sum with ψ; the second sum is called 'sharp-sharp' since it involves indicator functions in both λ j and in µ k . The smoothed sum N 1 ψ,H (λ) is technically simpler to work with and, perhaps surprisingly, often has better applications.…”