On the Quantum Reconstruction of the Riemann zeros

TL;DR

This paper explores a quantum Hamiltonian model with specific potentials that could realize the Riemann zeros as spectral points, potentially providing a quantum interpretation of the zeros.

Contribution

It introduces a perturbative approach to identify potentials related to the Riemann zeta function as a Jost function, linking spectral theory to the zeros.

Findings

Potentials related to the zeta function are identified for different values of sigma.

At sigma=1/2, potentials produce a smooth approximation to the Riemann zeros.

Existence of such potentials implies the zeros are embedded in a continuum spectrum.

Abstract

We discuss a possible spectral realization of the Riemann zeros based on the Hamiltonian $H = xp$ perturbed by a term that depends on two potentials, which are related to the Berry-Keating semiclassical constraints. We find perturbatively the potentials whose Jost function is given by the zeta function $\zeta(\sigma - i t)$ for $\sigma > 1$. For $\sigma = 1/2$ we find the potentials that yield the smooth approximation to the zeros. We show that the existence of potentials realizing the zeta function at $\sigma = 1/2$, as a Jost function, would imply that the Riemann zeros are point like spectrum embedded in the continuum, resolving in that way the emission/spectral interpretation.