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].
Highlights
Riemann’s outstanding hypothesis [1], that the nontrivial complex zeroes of the zeta-function ζ(s) must be of the form sn = 1/2 ± iλn, is one of most important open problems in pure mathematics
In [28], we studied a modified Dirac operator involving a potential related to the number counting function of zeta zeroes and left the Schroedinger operator case for a future project that we undertake in this work
Concluding Remarks The Bohr–Sommerfeld formula, in conjunction with the full-fledged Riemann–von Mangoldt counting formula for the number N (E) of zeroes in the critical strip, with an imaginary part greater than 0 and less than or equal to E, was used to construct a saw potential comprised of an infinite number of branches
Summary
Lagarias [17] studied the Schroedinger Operator with a Morse (exponential) potential on the right half-line and obtained information on the location of zeroes of the Whittaker function Wκ,μ(x) for fixed real parameters κ, x with x > 0, viewed as an entire function of the complex variable μ. In [28], we studied a modified Dirac operator involving a potential related to the number counting function of zeta zeroes and left the Schroedinger operator case for a future project that we undertake in this work
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More From: International Journal of Geometric Methods in Modern Physics
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