Canzani, Y;
Galkowski, J;
(2021)
Eigenfunction concentration via geodesic beams.
Journal für die reine und angewandte Mathematik (Crelles Journal)
, 2021
(775)
pp. 197-257.
10.1515/crelle-2020-0039.
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Abstract
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.
| 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 |
| Publisher version: | https://doi.org/10.1515/crelle-2020-0039 |
| Language: | English |
| Additional information: | This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions. |
| UCL classification: | UCL 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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