Hamiltonian with Energy Levels Corresponding to Riemann Zeros

A Hamiltonian with eigenenergy $E_n = \rho_n(1 - \rho_n)$ 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 paradigm and encode number-theoretic information into the Hamiltonian using modular forms.Although our construction does not resolve the Hilbert-P\'olya conjecture (since the eigenstates corresponding to $E_n$ are \emph{not} normalizable), it provides a novel physical perspective on the Riemann Hypothesis (RH). In particular, we propose a physical interpretation of RH, which could offer a potential pathway toward its proof.