Eigenfunction concentration via geodesic beams

Abstract

AbstractWe develop new techniques for studying concentration of Laplace eigenfunctionsϕλ{\phi_{\lambda}}as their frequency, λ, grows. The method consists of controllingϕλ⁢(x){\phi_{\lambda}(x)}by decomposingϕλ{\phi_{\lambda}}into a superposition of geodesic beams that run through the pointx. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower thanλ-12{\lambda^{-\frac{1}{2}}}. We controlϕλ⁢(x){\phi_{\lambda}(x)}by theL2{L^{2}}-mass ofϕλ{\phi_{\lambda}}on each geodesic tube and derive a purely dynamical statement through whichϕλ⁢(x){\phi_{\lambda}(x)}can be studied. In particular, we obtain estimates onϕλ⁢(x){\phi_{\lambda}(x)}by decomposing the set of geodesic tubes into those that are non-self-looping for timeTand those that are. This approach allows for quantitative improvements, in terms ofT, on the available bounds forL∞{L^{\infty}}-norms,Lp{L^{p}}-norms, pointwise Weyl laws, and averages over submanifolds.