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

Abstract

AbstractFor a full rank lattice $$\Lambda \subset \mathbb {R}^d$$ Λ ⊂ R d and $$\textbf{A}\in \mathbb {R}^d$$ A ∈ R d , consider $$N_{d,0;\Lambda ,\textbf{A}}(\Sigma ) = \# ([\Lambda +\textbf{A}] \cap \Sigma \mathbb {B}^d) = \# \{\textbf{k}\in \Lambda : |\textbf{k}+\textbf{A}| \le \Sigma \}$$ N d , 0 ; Λ , A ( Σ ) = # ( [ Λ + A ] ∩ Σ B d ) = # { k ∈ Λ : | k + A | ≤ Σ } . Consider the iterated integrals $$\begin{aligned} N_{d,k+1;\Lambda ,\textbf{A}}(\Sigma ) = \int _0^\Sigma N_{d,k;\Lambda ,\textbf{A}}(\sigma ) \,\textrm{d}\sigma , \end{aligned}$$ N d , k + 1 ; Λ , A ( Σ ) = ∫ 0 Σ N d , k ; Λ , A ( σ ) d σ , for $$k\in \mathbb {N}$$ k ∈ N . After an elementary derivation via the Poisson summation formula of the sharp large-$$\Sigma $$ Σ asymptotics of $$N_{3,k;\Lambda ,\textbf{A}}(\Sigma )$$ N 3 , k ; Λ , A ( Σ ) for $$k\ge 2$$ k ≥ 2 (these having an $$O(\Sigma )$$ O ( Σ ) error term), we discuss how they are encoded in the structure of the Fourier transform $$\mathcal {F}N_{3,k;\Lambda ,\textbf{A}}(\tau )$$ 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 d-torus. That the $$N_{3,k;\Lambda ,\textbf{A}}(\Sigma )$$ N 3 , k ; Λ , A ( Σ ) obey an asymptotic expansion to $$O(\Sigma ^2)$$ 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 half-wave trace at $$\tau =0$$ τ = 0 . The improvement to $$O(\Sigma )$$ O ( Σ ) for $$k\ge 2$$ k ≥ 2 follows from a bound on the growth rate of the half-wave trace at late times.