The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

Abstract

Inspired by the Hilbert–Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly nontrivial potential, and whose self-adjoint Hamiltonian operator has for its energy spectrum one which approaches the imaginary parts of the zeta zeroes only in the asymptotic (very large [Formula: see text]) region. The ordinates [Formula: see text] are the positive imaginary parts of the nontrivial zeta zeroes in the critical line :[Formula: see text]. The latter results are consistent with the validity of the Bohr–Sommerfeld semi-classical quantization condition. It is shown how one may modify the parameters which define the potential, and fine tune its values, such that the energy spectrum of the (modified) Hamiltonian matches not only the first two zeroes but the other consecutive zeroes. The highly nontrivial functional form of the potential is found via the Bohr–Sommerfeld quantization formula using the full-fledged Riemann–von Mangoldt counting formula (without any truncations) for the number [Formula: see text] of zeroes in the critical strip with imaginary part greater than [Formula: see text] and less than or equal to [Formula: see text].