On the Hamiltonian with Energy Levels Corresponding to Riemann Zeros

• URL: http://arxiv.org/abs/2505.21192v4
• Date: Wed, 06 Aug 2025 06:12:02 GMT
• Title: On the Hamiltonian with Energy Levels Corresponding to Riemann Zeros
• Authors: Xingpao Suo,
• Abstract summary: A Hamiltonian with eigenenergies $E_n = rho_n (1-rho_n) $ has been constructed.<n>We generalize the Berry-Keating's paradigm and encode number-theoretic information into the Hamiltonian through modular forms.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A Hamiltonian with eigenenergies $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.

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