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 $ζ(σ- i t)$ 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 spectrum embedded in the continuum, resolving in that way the emission/spectral interpretation.

26 pages, 5 figures, to appear in the Proceedings of the ``5th International Symposium on Quantum Theory and Symmetries'' held at University of Valladolid, Spain, 2007