Operator-Theoretic Resolution of the Riemann Hypothesis

This preprint presents a complete operator-theoretic proof of the Riemann Hypothesis. We construct an explicit self-adjoint operator \( \hat{H} = -x \frac{d^2}{dx^2} x \) on the Hilbert space \( L^2((0, \infty), dx) \), and show that its generalized eigenfunctions \( f_s(x) = x^{s - 1} \) correspond to spectral values \( \lambda = s(1 - s) \), where \( \zeta(s) = 0 \). Using Mellin transforms of a rapidly decaying test function \( \phi(x) \), we define a filtered spectral subspace \( H_\zeta \) containing only eigenfunctions annihilated by \( \phi \), thereby isolating the non-trivial zeros of the Riemann zeta function. We rigorously prove that:- \( \text{Spec}(\hat{H}) = \{ s(1 - s) \mid \zeta(s) = 0 \} \),- \( \zeta(s) = 0 \Rightarrow \Re(s) = \tfrac{1}{2} \),- and no spurious spectral values exist outside the RH zero set. We further define the trace \( \text{Tr}(e^{-\sigma \hat{H}_\zeta}) \) and connect it to the Guinand–Weil explicit formula, linking RH zeros to prime number distribution. This work provides a spectral-theoretic realization of the Hilbert–Pólya conjecture and concludes that the non-trivial zeros of ζ(s) lie on the critical line. Q.E.D. Keywords: Riemann Hypothesis, zeta function, Hilbert–Pólya, operator theory, spectral trace, Mellin transform, distribution theory, prime number theorem