Long time behavior of the half-wave trace and Weyl remainders

Abstract

Given a compact Riemannian manifold $(M,g)$, Chazarain, Hörmander, Duistermaat, and Guillemin study the half-wave trace ${\mathrm{HWT}}_{M,g}(\tau )\in {\mathcal{S}}^{\prime }({\mathbb{R}}_{\tau })$. From the asymptotics of the half-wave trace as $\tau \to 0$, Hörmander deduces the now standard remainder $O({\sigma }^{d-1})=O({\lambda }^{d/2-1/2})$ in Weyl's law, where $d=\dim M$. Given a dynamical assumption implying additional local regularity, Duistermaat and Guillemin improve this to $o({\sigma }^{d-1})$. By examining the Tauberian step in the argument, we show how a quantitative version \[N(σ) = Z(σ) + O(σ^{d-1}\mathcal{R}(σ)^{-1/2})\] of the Duistermaat-Guillemin result follows under slightly stronger hypotheses, these implying that the $(d-1)$-fold regularized half-wave trace \[\langle D_τ\rangle^{1-d} \operatorname{HWT}_{M,g}(τ)\] is in $L_{\mathrm{loc}}^{1,1}(\mathbb{R}\backslash \{0\})$. 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 $⟨D_{\tau }{⟩}^{1-d}{\tau }^{-1}{\mathrm{HWT}}_{M,g}(\tau )$ as measured in $L^{1,1}$. Per Duistermaat-Guillemin, this hypothesis is implied by geometric conditions that hold ``generically'' for $d\ge 3$. Thus, we clarify the relation between the error term in Weyl's law and the long time behavior of the half-wave trace.

arXiv comment22 pages. More general main theorem

References in corpus (1)

• Lattice points in large regions and related arithmetic functions: Recent developments in a very classic topic