Canzani, Yaiza;
Galkowski, Jeffrey;
(2020)
Weyl remainders: an application of geodesic beams.
arXiv.org: Ithaca (NY), USA.
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Abstract
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.
| Type: | Working / discussion paper |
|---|---|
| Title: | Weyl remainders: an application of geodesic beams |
| Open access status: | An open access version is available from UCL Discovery |
| Publisher version: | https://doi.org/10.48550/arXiv.2010.03969 |
| Language: | English |
| Additional information: | This version is the author manuscript. 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/10138601 |
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