Eigenfunction concentration via geodesic beams

We develop new techniques for studying concentration of Laplace eigenfunctions ϕλ as their frequency, λ, grows. The method consists of controlling ϕλ(x) by decomposing ϕλ into a superposition of geodesic beams that run through the point x. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than λ- 1 2 . We control ϕλ(x) by the L2-mass of ϕλ on each geodesic tube and derive a purely dynamical statement through which ϕλ(x) can be studied. In particular, we obtain estimates on ϕλ(x) by decomposing the set of geodesic tubes into those that are non-self-looping for time T and those that are. This approach allows for quantitative improvements, in terms of T, on the available bounds for L∞-norms, Lp-norms, pointwise Weyl laws, and averages over submanifolds

On the growth of eigenfunction averages: Microlocalization and geometry

Let (M,g) be a smooth, compact Riemannian manifold, and let {φ_{h]} be an L² -normalized sequence of Laplace eigenfunctions, -h²Δ_{g}φ_{h} = φ_{h}. Given a smooth submanifold H ⊂ M of codimension K ≥ 1, we find conditions on the pair ({φ_{h}, H) for which |∫_{H}φ_{h} δσ_{H}| = o (h\frac{1-k}{2}), h→0⁺. One such condition is that the set of conormal directions to H that are recurrent has measure 0. In particular, we show that the upper bound holds for any H if (M, g) is a surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages

Weyl remainders: an application of geodesic beams

We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold (M,g) of dimension n, let Πλ denote the kernel of the spectral projector for the Laplacian, 1[0,λ2](−Δg). Assuming only that the set of near periodic geodesics over W⊂M has small measure, we prove that as λ→∞ ∫WΠλ(x,x)dx=(2π)−nvolRn(B)volg(W)λn+O(λn−1logλ), where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector Πλ(x,y) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams

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

This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non-self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n. Keywords: Spectral projector, Pointwise Weyl Law, Scaling limits, Laplace eigenfunctions, Non-self-focal point