On the Hamiltonian with Energy Levels Corresponding to Riemann Zeros

Abstract: A Hamiltonian with eigenvalues $E_n = \rho_n(1-\rho_n) $ has been constructed, where $\rho_n $ denotes the $n-$th non-trivial zero of the Riemann zeta function. To construct such a Hamiltonian, we generalize the Berry-Keating's paradigm and encode number-theoretic information into the Hamiltonian through modular forms. Even though our construction does not resolve the Hilbert-P\'olya conjecture -- since the eigenstates corresponding to $E_n$ are \emph{not} normalizable states -- it offers a novel physical perspective on the Riemann Hypothesis(RH). Especially, we proposed a physical statement of RH, which may serve as a potential pathway toward its proof.