Hamiltonian with Energy Levels Corresponding to Riemann Zeros

2025/05/27 by Suo, Xingpao
#FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2505.21192

Abstract

A Hamiltonian with eigenenergy \( E_n = ρ_n(1 - ρ_n) \) has been constructed, where \( ρ_n \) 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-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.

Citations

• A proof of Newman's conjecture for the extended Selberg class

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