Towards a Numerical Realization of the Hilbert–Pólya Operator: A Partial Spectral Construction Approach

This article presents a partial spectral construction of a Hermitian operator whose spectrum aligns with the Riemann zeta non-trivial zeros, inspired by the Hilbert–Pólya conjecture. Using inverse spectral methods and the Gel'fand–Levitan approach, we numerically reconstruct a smooth potential V(x) and examine regularity, domain properties, and limitations of this model. While not a proof of the Riemann Hypothesis, the framework offers a constructive path toward operator-theoretic formulations of RH and encourages further analytical refinement.