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Eigenfunction concentration via geodesic beams

Canzani, Y; Galkowski, J; (2020) Eigenfunction concentration via geodesic beams. Journal fur die Reine und Angewandte Mathematik 10.1515/crelle-2020-0039. Green open access

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Abstract

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, Lp-norms, pointwise Weyl laws, and averages over submanifolds.

Type: Article
Title: Eigenfunction concentration via geodesic beams
Open access status: An open access version is available from UCL Discovery
DOI: 10.1515/crelle-2020-0039
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10110426
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