Prolate Leakage and Near-Zero Spectral Clusters of the Semilocal Weil Operator

Abstract

For $\lambda>1$ let $A_\lambda$ be the self-adjoint operator on $L^2([\lambda^{-1},\lambda],du/u)$ that represents the semilocal Weil quadratic form, including the archimedean place, all prime powers up to $\lambda^2$ and the pole terms; it has compact resolvent. We prove that for every integer $m\ge1$ there are explicit constants $\mathcal{K}_m$ and $\Lambda_m$ such that, for every real $\lambda\ge\Lambda_m$, the operator $A_\lambda$ has at least $m$ eigenvalues, counted with multiplicity, in $[-\delta,\delta]$, where $\delta=\mathcal{K}_m\lambda^5(1-\chi_{4m+4}(\lambda))^{1/2}$ and the $\chi_n(\lambda)$ are the eigenvalues of time and frequency limiting to $[-\lambda,\lambda]$. Known effective bounds for $1-\chi_n$ give $\delta\le\mathcal{K}_m'\lambda^{4m+17/2}e^{-\pi\lambda^2}$. Unlike upper bounds for the bottom of the spectrum, this localizes eigenvalues on both sides of zero, without any positivity assumption. The test vectors are images, under the summation map $f\mapsto u^{1/2}\sum_{n\ge1}f(nu)$, of combinations of prolate spheroidal functions with $f(0)=\hat f(0)=0$. The proof rests on three ingredients: an exact residual identity, derived from the explicit formula and Poisson summation, which expresses $A_\lambda$ on such images through the part of $\hat f$ outside $[-\lambda,\lambda]$; a transfer estimate bounding this residual by $\lambda^5$ times the leakage; and a lower bound for the summation map on blocks of prolate functions, uniform in $\lambda$. Consequently the number of eigenvalues of $A_\lambda$ in $[-e^{-(\pi-\epsilon)\lambda^2},e^{-(\pi-\epsilon)\lambda^2}]$ tends to infinity. No hypothesis on the zeros of the zeta function is used. Time flies, and I want to conduct more high-quality research.