On the quantum reconstruction of the Riemann zeros

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 ζ(σ − it) for σ > 1. For σ = 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 σ = 1/2, as a Jost function, would imply that the Riemann zeros are point-like spectra embedded in the continuum, resolving in that way the emission/spectral interpretation.