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

TL;DR

This paper derives sharp asymptotics for lattice point counts on the 3-torus, relates them to spectral Riesz means and Fourier analysis, and extends results to magnetic Schr"odinger operators on manifolds.

Contribution

It provides elementary derivations of asymptotics for iterated lattice point counts and connects these to spectral theory and Fourier transforms, extending to general magnetic Schr"odinger operators.

Findings

Asymptotics of lattice point counts with $O( ext{Sigma})$ error for $k e 1$

Connection between lattice counts and Fourier transform structure

Extension of asymptotic expansion results to all magnetic Schr"odinger operators on manifolds

Abstract

For a full rank lattice $\Lambda \subset \mathbb{R}^d$ and $\mathbf{A} \in \mathbb{R}^d$, consider $N_{d,0;\Lambda,\mathbf{A}}(\Sigma) = \# ([\Lambda+\mathbf{A}] \cap \Sigma \mathbb{B}^d) = \# \{\mathbf{k}\in \Lambda : |\mathbf{k}+\mathbf{A}| \leq \Sigma \}$. Consider the iterated integrals \[ N_{d,k+1;\Lambda,\mathbf{A}}(\Sigma) = \int_0^\Sigma N_{d,k;\Lambda,\mathbf{A}}(\sigma) \,\mathrm{d} \sigma, \] 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 $k\geq 2$ (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 H\"ormander's analysis of spectral Riesz means, as the iterated integrals above are weighted spectral Riesz means for the simplest magnetic Schr\"odinger 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 Schr\"odinger 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 $k\geq 2$ follows from a bound on the growth rate of the half-wave trace at late times.