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

Abstract

For a full rank lattice Lambda subset mathbb {R}^d and textbf{A}in mathbb {R}^d, consider N_{d,0;Lambda ,textbf{A}}(Sigma ) = # ([Lambda +textbf{A}] cap Sigma mathbb {B}^d) = # {textbf{k}in Lambda : |textbf{k}+textbf{A}| le Sigma }. Consider the iterated integrals Nd,k+1;Λ,A(Σ)=∫0ΣNd,k;Λ,A(σ)dσ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} N_{d,k+1;\\Lambda ,\ extbf{A}}(\\Sigma ) = \\int _0^\\Sigma N_{d,k;\\Lambda ,\ extbf{A}}(\\sigma ) \\,\ extrm{d}\\sigma , \\end{aligned}$$\\end{document}for kin mathbb {N}. After an elementary derivation via the Poisson summation formula of the sharp large-Sigma asymptotics of N_{3,k;Lambda ,textbf{A}}(Sigma ) for kge 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,k;Lambda ,textbf{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 d-torus. That the N_{3,k;Lambda ,textbf{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 kge 2 follows from a bound on the growth rate of the half-wave trace at late times.