A Hamiltonian with eigenenergy $E_n={\rho }_n(1-{\rho }_n)$ has been constructed, where ${\rho }_n$ denotes the $n\mathrm{th}$ nontrivial 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-Pólya conjecture (since the eigenstates corresponding to $E_n$ are 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.

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• Nonrelativistic wave equations
• Schroedinger equation