Canzani, Yaiza;
Galkowski, Jeffrey;
(2023)
Weyl remainders: an application of geodesic beams.
Inventiones Mathematicae
, 232
pp. 1195-1272.
10.1007/s00222-023-01178-5.
Preview |
Text
Galkowski_s00222-023-01178-5.pdf Download (1MB) | Preview |
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 $\Pi_\lambda$ denote the kernel of the spectral projector for the Laplacian, $\mathbb{1}_{[0,\lambda^2]}(-\Delta_g)$. Assuming only that the set of near periodic geodesics over $W\subset M$ has small measure, we prove that as $\lambda \to \infty$ $$ \int_{W} \Pi_\lambda(x,x)dx=(2\pi)^{-n}\text{vol}_{\mathbb{R}^n}(B)\text{vol}_g(W)\,\lambda^n+O\Big(\frac{\lambda^{n-1}}{\log \lambda }\Big),$$ 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 $\Pi_\lambda(x,y)$ 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: | Article |
|---|---|
| Title: | Weyl remainders: an application of geodesic beams |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00222-023-01178-5 |
| Publisher version: | https://doi.org/10.1007/s00222-023-01178-5 |
| Language: | English |
| Additional information: | This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
| 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-pp.ucl.ac.uk/id/eprint/10163744 |
Archive Staff Only
 |
View Item |
