Prolate spheroidal operator and Zeta

TL;DR

This paper uncovers a new spectral property of the prolate spheroidal operator related to the zeros of the Riemann zeta function, linking operator theory with number theory and spectral analysis.

Contribution

It reveals that the restriction of the operator to the complement of a certain interval admits negative eigenvalues mirroring the zeros of the Riemann zeta function, and constructs isospectral Dirac operators.

Findings

Negative eigenvalues reproduce zeta zeros' ultraviolet behavior.

Eigenfunctions belong to the Sonin space.

Constructs isospectral Dirac operators with zeta-like spectra.

Abstract

In this paper we describe a remarkable new property of the self-adjoint extension W of the prolate spheroidal operator introduced in \cite{college98},\cite{CMbook}. The restriction of this operator to the interval J whose characteristic function commutes with it is well known, has discrete positive spectrum and is well understood. What we have discovered is that the restriction of W to the complement of J admits (besides a replica of the above positive spectrum) negative eigenvalues whose ultraviolet behavior reproduce that of the squares of zeros of the Riemann zeta function. Furthermore, their corresponding eigenfunctions belong to the Sonin space. This feature fits with the proof \cite{weilpos} of Weil's positivity at the archimedean place, which uses the compression of the scaling action to the Sonin space. As a byproduct we construct an isospectral family of Dirac operators whose spectra have the same ultraviolet behavior as the zeros of the Riemann zeta function.