2021/09/21 by Ethan Sussman, Sussman, Ethan
Mathematics · #11M45 #35P20 #42axx #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Analytic Number Theory Research #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2109.09926
openalex publication_date 2021/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a compact Riemannian manifold (M,g), Chazarain, Hörmander, Duistermaat, and Guillemin study the half-wave trace HWTM,g(τ) ∈ \mathscrS'(ℝτ). From the asymptotics of the half-wave trace as τ→ 0, Hörmander deduces the now standard remainder \smashO(σd-1) = O(λd/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(σd-1). By examining the Tauberian step in the argument, we show how a quantitative version N(σ) = Z(σ) + O(σd-1R(σ)-1/2) of the Duistermaat-Guillemin result follows under slightly stronger hypotheses, these implying that the (d-1)-fold regularized half-wave trace ⟨ Dτ⟩1-d HWTM,g(τ) is in \smashL1,1loc(ℝ\backslash \0\). Here Z(σ)∈ ℝ[σ] is a polynomial and R(σ):ℝ+→ ℝ+ is an (M,g)-dependent nondecreasing function with limσ→∞ R(σ)=∞, specified in terms of the growth rate of ⟨ Dτ⟩1-d τ-1HWTM,g(τ) as measured in L1,1. Per Duistermaat-Guillemin, this hypothesis is implied by geometric conditions that hold ``generically'' for d≥ 3. Thus, we clarify the relation between the error term in Weyl's law and the long time behavior of the half-wave trace.