On self-adjointness of Poisson summation

TL;DR

This paper demonstrates that a specific combination of well-known operators related to Poisson summation is self-adjoint and connected to the Riemann zeta function, with implications for the Hilbert-Pólya conjecture and related mathematical structures.

Contribution

It introduces a new self-adjoint operator constructed from classical operators, linking Poisson summation to the Riemann zeta function and exploring related mathematical frameworks.

Findings

The operator combination is proven to be self-adjoint.

Connections to the Riemann zeta function and Weil's positivity criteria are established.

Implications for the Hilbert-Pólya conjecture are discussed.

Abstract

We show that a combination of well-known operators, namely $\I{\tau}\circ{H}\circ\Ps$ 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-P\'olya conjecture, discuss why the Hermite-Biehler theorem, uncertainty relations and cohomologies are interesting in our scenario.