Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law

Abstract

Abstract. Let (M, g) be a compact smooth Riemannian manifold. We obtain new off-diagonal estimates as λ → ∞ for the remainder in the pointwise Weyl Law for the kernel of the spectral projector of the Laplacian onto functions with frequency at most λ. A corollary is that, when rescaled around a non self-focal point, the kernel of the spectral projector onto the frequency interval (λ, λ + 1] has a universal scaling limit as λ → ∞ (depending only on the dimension of M ). Our results also imply that if M has no conjuage points, then immersions of M into Euclidean space by an orthonormal basis of eigenfunctions with frequencies in (λ, λ + 1] are embeddings for all λ sufficiently large.

Volume distribution of nodal domains of random band-limited functions

“…The following semiclassical approximation, universal w.r.t. M holds (see [8, Section 2.1] with case α = 1 due to Canzani-Hanin [3,4]): where A α = {w : α ≤ |w| ≤ 1} and in the case α = 1 the n-dimensional measure dξ is replaced by (n − 1)-dimensional surface measure on the unit sphere.…”

$$C^\infty $$ C ∞ Scaling Asymptotics for the Spectral Projector of the Laplacian

“…That result was in turn was based on the work of Sogge-Zelditch [18,19], who studied j = k = 0 and x = y. This last situation was also studied (independently and significantly before [4,18,19]) by Safarov in [12] (cf [13]) using a somewhat different method. The case j = k = 1 and x = y is essentially Proposition 2.3 in [20].…”

“…Our main technical result, Theorem 2, shows that the remainder estimate (5) for R(x, y, λ) can be improved from O(λ n−1+j+k ) to o(λ n−1+j+k ) under the assumption that x and y are near a non self-focal point (defined below). This paper is a continuation of [4] where the authors proved Theorem 2 for j = k = 0. An application of our improved remainder estimates is Theorem 1, which shows that we can compute the scaling limit of E (λ,λ+1] (x, y) and its derivatives near a non self-focal point as λ → ∞.…”

“…The integral in (4) is over the cotangent fiber T * y M and the integration measure is the quotient of the symplectic form dξ ∧dy by the Riemannian volume form dv g = |g y |dy. In Hörmander's original theorem, the phase function exp −1 y (x), ξ is replaced by any so-called adapted phase function and one still obtains that sup dg(x,y)<ε…”

Shrinking Scale Equidistribution for Monochromatic Random Waves on Compact Manifolds

“…Using the wave equation in this way may break down when T d(x, y) is unbounded. For larger distances we instead appeal to the results of Canzani-Hanin [7]. Their Theorem 2 improves the O(T n−1 ) error term in Hörmander's estimate for K(x, y) to o(T n−1 ), assuming x, y are in a ball B r (z) of radius r → 0 arbitrarily slowly around some non-self-focal point z.…”