Eigenfunction concentration via geodesic beams

2019/03/20 by Canzani, Yaiza, Galkowski, Jeffrey · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1903.08461

Abstract

In this article we develop new techniques for studying concentration of Laplace eigenfunctions ϕ_λ as their frequency, λ, grows. The method consists of controlling ϕ_λ(x) by decomposing ϕ_λ 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 ϕ_λ(x) by the L²-mass of ϕ_λ on each geodesic tube and derive a purely dynamical statement through which ϕ_λ(x) can be studied. In particular, we obtain estimates on ϕ_λ(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, L^p norms, pointwise Weyl laws, and averages over submanifolds.

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