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

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 generate it; specifically, eigenfunctions for n greater than zero do not belong to the completed space. This finding addresses a core goal of the Hilbert-Pólya conjecture, which seeks a self-adjoint operator whose spectrum matches the imaginary parts of the nontrivial zeros of the Riemann zeta function. The analysis also demonstrates that no bounded sandwich operator is boundedly invertible, a spectral statement extending beyond the BBM problem itself, and that the transported symmetric operator possesses deficiency indices (1,1) with an adjoint exhibiting infinite multiplicity at every real eigenvalue.

BBM Candidate Metric: Non-Coercivity and Completion

The pursuit of a self-adjoint operator whose spectrum mirrors the Riemann zeta function’s non-trivial zeros, the Hilbert-Pólya conjecture, continues to drive innovative approaches in mathematical physics. Recent analysis of the Bender-Brody-Müller (BBM) Hamiltonian, proposed as a potential candidate, reveals fundamental limitations in constructing a suitable Hilbert space for this operator, stemming from the properties of its associated metric. The team employed a Fourier representation to dissect the BBM form, revealing that the metric assigns arbitrarily small length to normalized states concentrated near the zeros of the boundary symbol, indicating a weakening of the standard Hilbert space topology. This degeneracy is not a matter of ill-defined eigenvectors, but rather a fundamental property of the form itself; the metric “differentiates against a unit shift, so slowly varying wave packets—Fourier-concentrated near the zeros of —have arbitrarily small metric length.” Crucially, the study establishes that, as far as they have determined, these eigenfunctions do not belong to the completed space.

The analysis shows that the original BBM boundary-condition/eigenfunction mechanism cannot produce point-spectrum Riemann-zero states in this metric completion. The free extension of this operator, however, remains purely continuous. The work clarifies that the BBM similarity structure, candidate metric, and standard completion procedure, while mathematically sound, ultimately lead to a completed space that excludes the desired eigenfunctions, offering a precise structural constraint on potential Hilbert-Pólya constructions.

This work does not directly solve the century-old mathematical problem, but clarifies fundamental limitations within the BBM framework itself, and reveals broader constraints on constructing Hilbert spaces for such problems. The core of Liu’s investigation centers on the Hilbert completion induced by the BBM Hamiltonian’s candidate metric, a process of extending the initial mathematical space to accommodate the operator’s properties. Further complicating matters, the analysis demonstrates that no bounded sandwich operator is boundedly invertible. The key conclusion concerning the candidate eigenfunctions is that they do not belong to the completed space. This is not merely a problem for the BBM model; the analysis yields two spectral statements of interest beyond the BBM problem. The work highlights that even with a carefully constructed formal Hamiltonian, the choice of Hilbert space and metric can dramatically impact the validity and physical interpretation of the results, and that “the physical metric topology selected by the BBM similarity transform is itself noncoercive.”

Central to these findings are the deficiency indices of the transported symmetric operator, determined to be (1,1).