Half-waves and spectral Riesz means on the 3-torus

Abstract

For a full rank lattice $\Lambda \subset {\mathbb{R}}^d$ and $\mathbf{A}\in {\mathbb{R}}^d$, consider $N_{d,0;\Lambda ,\mathbf{A}}(\Sigma )=\#([\Lambda +\mathbf{A}]\cap \Sigma {\mathbb{B}}^d)=\#\{\mathbf{k}\in \Lambda :|\mathbf{k}+\mathbf{A}|\le \Sigma \}$. Consider the iterated integrals \[ N_{d,k+1;Λ,\mathbf{A}}(Σ) = \int_0^ΣN_{d,k;Λ,\mathbf{A}}(σ) \,\mathrm{d} σ, \] for $k\in \mathbb{N}$. After an elementary derivation via the Poisson summation formula of the sharp large-$\Sigma$ asymptotics of $N_{3,k;\Lambda ,\mathbf{A}}(\Sigma )$ for $k\ge 2$ (these having an $O(\Sigma )$ error term), we discuss how they are encoded in the structure of the Fourier transform $\mathcal{F}{N}_{3;\Lambda ,\mathbf{A}}(\tau )$. The analysis is related to Hörmander's analysis of spectral Riesz means, as the iterated integrals above are weighted spectral Riesz means for the simplest magnetic Schrödinger operator on the flat $3$-torus. That the $N_{3,k;\Lambda ,\mathbf{A}}(\Sigma )$ obey an asymptotic expansion to $O({\Sigma }^2)$ is a special case of a general result holding for all magnetic Schrödinger operators on all manifolds, and the subleading polynomial corrections can be identified in terms of the Laurent series of the half-wave trace at $\tau =0$. The improvement to $O(\Sigma )$ for $k\ge 2$ follows from a bound on the growth rate of the half-wave trace at late times.

arXiv comment27 pages, 1 figure. To appear in Anal. Math. Phys