Metric completion of the Bender--Brody--M\"uller Hamiltonian: dilation spectrum and missing eigenstates

arXiv:2607.19067v1 Announce Type: cross Abstract: The Bender--Brody--M\"uller (BBM) Hamiltonian was proposed as a non-Hermitian Hilbert--P\'olya operator. We analyze the Hilbert completion induced, on the standard half-line $L^2$ core, by BBM's candidate metric $\hat\eta=\sin^2(\hat p/2)=\Delta^\dagger\Delta/4$. The form $\eta_0=\Delta^\dagger\Delta$ is positive with trivial kernel but is not coercive. Completing $C_c^\infty(0,\infty)$ in the norm $\|\psi\|_{\eta_0}=\|\Delta\psi\|$ gives a Hilbert space canonically unitarily equivalent to $L^2(\mathbb R_+)$. 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 $\Delta^\dagger h(D)\Delta$ is boundedly invertible. Second, the transported symmetric operator has deficiency indices $(\infty,\infty)$ 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: $\Delta\psi_z=x^{-z}$, so for $\operatorname{Re}z=1/2$ 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 $L^2$-based metric completion.

Abstract: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 $L^2$ core, by BBM's candidate metric $\hat\eta=\sin^2(\hat p/2)=\Delta^\dagger\Delta/4$. The form $\eta_0=\Delta^\dagger\Delta$ is positive with trivial kernel but is not coercive. Completing $C_c^\infty(0,\infty)$ in the norm $\|\psi\|_{\eta_0}=\|\Delta\psi\|$ gives a Hilbert space canonically unitarily equivalent to $L^2(\mathbb R_+)$. 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 $\Delta^\dagger h(D)\Delta$ is boundedly invertible. Second, the transported symmetric operator has deficiency indices $(\infty,\infty)$ 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: $\Delta\psi_z=x^{-z}$, so for $\operatorname{Re}z=1/2$ 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 $L^2$-based metric completion.

Mathematical Physics (math-ph); Spectral Theory (math.SP); Quantum Physics (quant-ph)

arXiv:2607.19067 [math-ph]

(or arXiv:2607.19067v1 [math-ph] for this version)

https://doi.org/10.48550/arXiv.2607.19067 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Metric completion of the Bender--Brody--M\"uller Hamiltonian: dilation spectrum and missing eigenstates

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