A Nonlocal Spectral Operator Model Reflecting Zeta Function Structure

We propose a novel spectral operator framework inspired by the Hilbert–Pólya conjecture, aiming to model the nontrivial zeros of the Riemann zeta function as zero-energy eigenstates of a self-adjoint fractional Hamiltonian. Specifically, we define an operator of the form H := (−∆)γ + V (t), where γ ∈ (0, 1], (−∆)γ is a fractional Laplacian acting on a weighted Hilbert space Hζ = L 2 (R, µ(t) dt) with weight µ(t) = (1 + |t| 2 ) −α , and the potential V (t) := ℜ ζ ′ ζ 1 2 + it encodes the analytic structure of the Riemann zeta function. We establish essential self-adjointness of H via the Kato–Rellich theorem, construct a variational action functional whose critical points satisfy a nonlocal Schrödingertype equation, and demonstrate that the operator’s eigenvalue flow exhibits alignment with known nontrivial zeta zeros. Numerical simulations further reveal Gaussian Unitary Ensemble (GUE) statistics in level spacings, supporting connections to random matrix theory. While this formulation does not constitute a proof of the Riemann Hypothesis, it provides a rigorous, physically motivated, and computationally tractable operator model whose spectral structure reflects key properties of the zeta function. We offer this framework as a candidate system for further analytical and numerical investigation of the zeta spectrum from a spectral-theoretic and nonlocal dynamics perspective