On self-adjointness of Poisson summation

Abstract

We show that a combination of well-known operators, namely $\Iτ\circ{H}\circ\Ps$ is self-adjoint and {\em ad-hoc} related to the $\zeta$ function. Here $\tau$ is an involution appearing in Weil's positivity criteria needed for hermitrization, $H$ a regularization operator introduced by Connes \cite{Co2} and $\Ps$ essentially Poisson summation. We elaborate on the Hilbert-Pólya conjecture, discuss why the Hermite-Biehler theorem, uncertainty relations and cohomologies are interesting in our scenario.

Fixed some typos. Some details added