Raymond Aschheim, Carlos Castro Perelman, Klee Irwin (2016)

Inspired by the Hilbert-Polya proposal to prove the Riemann Hypothesis we have studied the Schroedinger QM equation involving a highly non-trivial 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 N) region. The ordinates $\lambda_n$ are the positive imaginary parts of the nontrivial zeta zeros in the critical line :  sn = 1/2 + i$\lambda_n$ . 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 non-trivial 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 N (E) of zeroes in the critical strip with imaginary part greater than 0 and less than or equal to E.