On the Existence of the Hilbert-Pólya Hamiltonian

Abstract: We demonstrate the existence of a self-adjoint Hamiltonian whose eigenvalues are of the form $i(1/2 - \rho_s)$, where $\rho_s$ denote the \textit{simple} nontrivial zeros of the Riemann zeta function. Consequently, all such zeros lie on the critical line, marking a significant step toward a proof of the Riemann Hypothesis. Moreover, our results suggest that all nontrivial Riemann zeros are simple.