On the Quantum Reconstruction of the Riemann zeros

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 $\zeta (\sigma -it)$ for $\sigma >1$. For $\sigma =1\mathrm{/}2$ we find the potentials that yield the smooth approximation to the zeros. We show that the existence of potentials realizing the zeta function at $\sigma =1\mathrm{/}2$, 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.