On the Quantum Reconstruction of the Riemann zeros

Details On the Quantum Reconstruction of the Riemann zeros On the Quantum Reconstruction of the Riemann zeros

2007-11-07

arxiv.org/abs/0711.1063v1

J.Phys.A41:304041,2008

Physics

Mathematical Physics

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

Scientific paper

10.1088/1751-8113/41/30/304041

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.

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