Long time behavior of the halfwave trace and Weyl remainders
Abstract
Given a compact Riemannian manifold $(M,g)$, Chazarain, Hörmander, Duistermaat, and Guillemin study the halfwave trace $\operatorname{HWT}_{M,g}(\tau) \in \mathscr{S}'(\mathbb{R}_\tau)$. From the asymptotics of the halfwave trace as $\tau\to 0$, Hörmander deduces the now standard remainder $\smash{O(\sigma^{d1}) = O(\lambda^{d/21/2})}$ in Weyl's law, where $d=\dim M$. Given a dynamical assumption implying additional local regularity, Duistermaat and Guillemin improve this to $o(\sigma^{d1})$. By examining the Tauberian step in the argument, we show how a quantitative version \[N(\sigma) = Z(\sigma) + O(\sigma^{d1}\mathcal{R}(\sigma)^{1/2})\] of the DuistermaatGuillemin result follows under slightly stronger hypotheses, these implying that the $(d1)$fold regularized halfwave trace \[\langle D_\tau \rangle^{1d} \operatorname{HWT}_{M,g}(\tau)\] is in $\smash{L^{1,1}_\mathrm{loc}(\mathbb{R}\backslash \{0\})}$. Here $Z(\sigma)\in \mathbb{R}[\sigma]$ is a polynomial and $\mathcal{R}(\sigma):\mathbb{R}^+\to \mathbb{R}^+$ is an $(M,g)$dependent nondecreasing function with $\lim_{\sigma\to\infty} \mathcal{R}(\sigma)=\infty$, specified in terms of the growth rate of $\langle D_\tau \rangle^{1d} \tau^{1}\operatorname{HWT}_{M,g}(\tau)$ as measured in $L^{1,1}$. Per DuistermaatGuillemin, this hypothesis is implied by geometric conditions that hold ``generically'' for $d\geq 3$. Thus, we clarify the relation between the error term in Weyl's law and the long time behavior of the halfwave trace.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 DOI:
 10.48550/arXiv.2109.09926
 arXiv:
 arXiv:2109.09926
 Bibcode:
 2021arXiv210909926S
 Keywords:

 Mathematics  Spectral Theory;
 Mathematics  Analysis of PDEs;
 11M45;
 35P20;
 42axx
 EPrint:
 22 pages. More general main theorem