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This research investigates whether the Bender–Brody–Müller (BBM) Hamiltonian can serve as a self-adjoint operator for the Hilbert–Pólya conjecture, which links spectral theory to the Riemann hypothesis. The analysis concludes that while a completed metric space exists, it fails to support the specific eigenfunctions required by the original boundary-condition mechanism proposed in 2013. Consequently, this work delineates fundamental mathematical constraints on using BBM as a candidate operator for modeling nontrivial zeta zeros within standard quantum mechanical frameworks.

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Hilbert–Pólya Finds Missing Eigenstates in BBM Hamiltonian’s Metric Completion

Quantum ZeitgeistAugust 13, 20263 min read

Analyzing the Bender–Brody–Müller (BBM) Hamiltonian as a Hilbert–Pólya operator, this work finds that certain candidate eigenfunctions do not exist in its completed space.

Kejun Liu of Soochow University has completed a detailed analysis of the Hilbert space structure underlying the Bender, Brody, Müller (BBM) Hamiltonian, originally proposed as a way to model the Riemann hypothesis. The work reveals that while a completed Hilbert space can be constructed using BBM’s candidate metric, it fundamentally limits the original boundary-condition/eigenfunction mechanism intended to…

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