2026/07/21 by Kejun Liu
#math-ph #math.MP #math.SP #quant-ph
The Bender--Brody--Müller (BBM) Hamiltonian was proposed as a non-Hermitian Hilbert--Pólya operator. We analyze the Hilbert completion induced, on the standard half-line L2 core, by BBM's candidate metric η=sin2( p/2)=Δ^†Δ/4. The form η0=Δ^†Δ is positive with trivial kernel but is not coercive. Completing Cc^∞(0,∞) in the norm ‖ψ‖η0=‖Δψ‖ gives a Hilbert space canonically unitarily equivalent to L2(\mathbb R+). Its free self-adjoint realization is the dilation generator, with simple, purely absolutely continuous spectrum \mathbb R. The analysis yields two spectral statements of interest beyond the BBM problem. First, no bounded sandwich Δ^† h(D)Δ is boundedly invertible. Second, the transported symmetric operator has deficiency indices (∞,∞) and an adjoint with every real point as an eigenvalue of infinite multiplicity, while its free extension is purely continuous. The realization-independent BBM conclusion concerns the candidate eigenfunctions: Δψz=x-z, so for Rez=1/2 they do not belong to the completed space. Thus the original BBM boundary-condition/eigenfunction mechanism cannot produce point-spectrum Riemann-zero states in this L2-based metric completion.