Successive generation of nontrivial Riemann zeros from a Wu-Sprung type potential

Abstract

A series of numerical experiments are performed, where a symmetric potential is generated for the 1D time-independent Schr\"odinger equation, with an eigenspectrum that matches the imaginary part of the first nontrivial zeros of the Riemann Zeta Function. The potential is generated as a series of correction functions, where the starting point is a potential that matches the smooth Riemann -- von Mangoldt approximation. It is found that the correction functions display a clear pattern that can be explained in simple terms, almost entirely dependent on the approximation error in the Riemann -- von Mangoldt formula. This also provides an explanation for the fractal pattern in the potential that was observed by Wu and Sprung.