On some bound and scattering states associated with the cosine kernel

TL;DR

This paper constructs self-adjoint operators with scattering states and bound states whose eigenvalue distribution mimics the zeros of the Riemann zeta function, linking spectral theory with number theory.

Contribution

It introduces a method to create operators with spectral properties reflecting the distribution of Riemann zeta zeros, extending recent integro-differential approaches.

Findings

Operators with scattering states forming a continuum

Bound states' eigenvalues match the asymptotic density of zeta zeros

Connection between spectral theory and the zeros of the Riemann zeta function

Abstract

It is explained how to provide self-adjoint operators having scattering states forming a multiplicity one continuum and bound states whose corresponding eigenvalues have an asymptotic density equivalent to the one of the zeros of the Riemann zeta function. It is shown how this can be put into an integro-differential form of a type recently considered by Sierra.