Halfwaves and spectral Riesz means on the 3torus
Abstract
For a full rank lattice Λ ⊂R^{d} and A ∈R^{d} , consider N_{d ,0 ;Λ ,A}(Σ ) =#([Λ +A ] ∩Σ B^{d}) =#{k ∈Λ :k +A ≤Σ } . Consider the iterated integrals N_{d ,k +1 ;Λ ,A}(Σ ) =∫0_{Σ}N_{d ,k ;Λ ,A}(σ ) dσ , for k ∈N . After an elementary derivation via the Poisson summation formula of the sharp largeΣ asymptotics of N_{3 ,k ;Λ ,A}(Σ ) for k ≥2 (these having an O (Σ ) error term), we discuss how they are encoded in the structure of the Fourier transform F N_{3 ,k ;Λ ,A}(τ ) . 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 dtorus. That the N_{3 ,k ;Λ ,A}(Σ ) obey an asymptotic expansion to O (Σ^{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 halfwave trace at τ =0 . The improvement to O (Σ ) for k ≥2 follows from a bound on the growth rate of the halfwave trace at late times.
 Publication:

Analysis and Mathematical Physics
 Pub Date:
 December 2022
 DOI:
 10.1007/s1332402200737y
 arXiv:
 arXiv:2109.10860
 Bibcode:
 2022AnMP...12..145F
 Keywords:

 35P20;
 11Lxx;
 42axx;
 Mathematics  Number Theory;
 Mathematics  Spectral Theory;
 35P20;
 11Lxx;
 42axx
 EPrint:
 27 pages, 1 figure. To appear in Anal. Math. Phys