Weyl remainders: an application of geodesic beams
Weyl remainders: an application of geodesic beams
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韦尔余数：测地梁的应用
DOI：
10.1007/s00222-023-01178-5
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发表时间：
2023
影响因子：
3.1
通讯作者：
Galkowski, Jeffrey
中科院分区：
文献类型：
--
作者：
Canzani, Yaiza;Galkowski, Jeffrey
We obtain newquantitativeestimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold (M,g) of dimensionn, letdenote the kernel of the spectral projector for the Laplacian,. Assumingonlythat the set of near periodic geodesics overhas small measure, we prove that as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$\end{document}whereBis the unit ball. One consequence of this result is that the improved remainder holds onallproduct 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 projectorunder the assumption that the set of geodesics that pass near bothxandyhas 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.
We obtain newquantitativeestimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold (M,g) of dimensionn, letdenote the kernel of the spectral projector for the Laplacian,. Assumingonlythat the set of near periodic geodesics overhas small measure, we prove that as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$\end{document}whereBis the unit ball. One consequence of this result is that the improved remainder holds onallproduct 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 projectorunder the assumption that the set of geodesics that pass near bothxandyhas 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.
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