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On Weyl’s asymptotics and remainder term for the orthogonal and unitary groups

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posted on 2023-06-09, 07:11 authored by Charles Morris, Ali TaheriAli Taheri
We examine the asymptotics of the spectral counting function of a compact Riemannian manifold by Avakumovic (Math Z 65:327–344, [1]) and Hörmander (Acta Math 121:193–218, [15]) and show that for the scale of orthogonal and unitary groups SO(N), SU(N), U(N) and Spin(N) it is not sharp. While for negative sectional curvature improvements are possible and known, cf. e.g., Duistermaat and Guillemin (Invent Math 29:39–79, [7]), here, we give sharp and contrasting examples in the positive Ricci curvature case [non-negative for U(N)]. Furthermore here the improvements are sharp and quantitative relating to the dimension and rank of the group. We discuss the implications of these results on the closely related problem of closed geodesics and the length spectrum.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Journal of Fourier Analysis and Applications

ISSN

1069-5869

Publisher

Springer Verlag

Issue

1

Volume

24

Page range

184-209

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Analysis and Partial Differential Equations Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2017-07-13

First Open Access (FOA) Date

2018-02-15

First Compliant Deposit (FCD) Date

2017-07-13

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