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

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Pith's one-line read The BBM Hamiltonian's own metric completion excludes its proposed zeta-zero eigenfunctions.

desk verdict A clean, well-scoped negative result: BBM's eigenfunctions do not survive the metric completion, with a solid supplement and only the expected scope caveat. read the letter →

arxiv 2607.19067 v1 pith:NUH663PF submitted 2026-07-21 math-ph math.MPmath.SPquant-ph

Kejun Liu This is my paper · ORCID
classification math-phmath.MPmath.SPquant-ph MSC 81Q1247B2511M26 PACS 03.65.-w
keywords BBMHamiltonianmetriccompletionpseudo-Hermitianquantummechanicsdilationgeneratornon-coercivedeficiencyindiceszeta-functionzerospurelycontinuousspectrum
topics The Riemann Hypothesis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks what Hilbert space the BBM Hamiltonian's candidate metric actually generates, and whether the proposed eigenfunctions survive that completion. It shows that completing the standard half-line test functions in the norm $\|\psi\|_{\eta_0}=\|\Delta\psi\|$ produces a Hilbert space canonically unitarily equivalent to $L^2(\mathbb R_+)$, with the free transported Hamiltonian conjugate to the dilation generator and purely absolutely continuous spectrum $\mathbb R$. The candidate eigenfunctions fail to survive: for $\mathrm{Re}\,z=1/2$ the shift difference $(\Delta\psi_z)(x)=x^{-z}$ is not square-integrable, so these functions are not vectors of the completed space. The paper concludes that no self-adjoint realization in this $L^2$-based metric completion can use the original BBM eigenfunctions and boundary condition to produce zeta-zero point-spectrum states.

What carries the argument

The central object is the positive but non-coercive quadratic form $\eta_0=\Delta^\dagger\Delta=2I-\hat S-\hat S^\dagger$, the half-line realization of BBM's proposed metric $\sin^2(\hat p/2)$. Its boundary symbol $\eta_0(k)=2(1-\cos k)$ vanishes at $k\in2\pi\mathbb Z$, creating soft directions but no null vectors. The argument runs through the completion of $C_c^\infty(0,\infty)$ in the norm $\|\psi\|_{\eta_0}=\|\Delta\psi\|_{L^2}$; the shift difference $\Delta$ extends to a unitary map onto $L^2(\mathbb R_+)$, transporting the formal Hamiltonian to the dilation generator $D=-i(2x\partial_x+1)$. The decisive identity is the shift-difference relation $(\Delta\psi_z)(x)=x^{-z}$, which puts the candidate eigenfunctions outside the completed space when $\mathrm{Re}\,z=1/2$. A separate mechanism is the no-go for bounded sandwiches: broad states $\psi_n(x)=n^{-1/2}\chi(x/n)$ have $\|\Delta\psi_n\|\to0$, so $\Delta^\dagger h(D)\Delta$ cannot be boundedly invertible for any bounded $h$.

What would settle it

Exhibit one self-adjoint extension of the transported symmetric operator $T$ in the $\eta_0$-completion that has $\psi_{1/2+i\gamma}$ as an eigenvector with eigenvalue $-2\gamma$, or equivalently find an $\eta_0$-Cauchy sequence of test functions whose limit represents that function. The direct calculation to check is whether any element of the completion can have shift difference $x^{-z}$; since $\int_1^\infty x^{-1}\,dx=\infty$, such a representative cannot exist.

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Extended reading notes

Core claim

The central claim is that the BBM similarity structure fixes a metric topology that is fatal to the original Hilbert–Pólya mechanism, even though it supports many operator realizations. The positive form $\eta_0=\Delta^\dagger\Delta=2I-\hat S-\hat S^\dagger$ is not coercive: normalized Cauchy-kernel probes concentrated near $k=2\pi$ have metric length $\langle\phi_\varepsilon,\eta_0\phi_\varepsilon\rangle=2(1-e^{-\varepsilon})\to0$. Completing $C_c^\infty(0,\infty)$ in $\|\Delta\psi\|$ gives a Hilbert space canonically unitarily equivalent to $L^2(\mathbb R_+)$ via $\Phi=\Delta\psi$; the free self-adjoint realization is the dilation generator $D$, whose spectrum is $\mathbb R$, simple and purely absolutely continuous. Two statements hold independently of zeta: no bounded sandwich $\Delta^\dagger h(D)\Delta$ is boundedly invertible, and the transported symmetric operator $T=D|_{\Delta D_0}$ has deficiency indices $(\infty,\infty)$, with the adjoint having every real point as an eigenvalue of infinite multiplicity while the free extension is purely continuous. Because $(\Delta\psi_z)(x)=x^{-z}$ on $x>1$, the BBM eigenfunctions on the critical line lie outside the completion, so every self-adjoint extension in this space fails to realize the proposed zeta-zero eigenfunction/boundary-condition mechanism.

Load-bearing premise

The load-bearing premise is that the physical state space is the Hilbert space obtained by completing smooth compactly supported half-line functions in the $\eta_0$ norm; if one instead adopts a more permissive topology such as a rigged Hilbert space, a distributional space, or an unbounded-metric formulation, the BBM eigenfunctions could in principle survive.

Editorial extensions

If this is right

  • In the $\eta_0$-completion, the free transported BBM Hamiltonian is unitarily equivalent to the dilation generator and has purely absolutely continuous spectrum $\mathbb R$, with no isolated levels.
  • The formal BBM operator has infinitely many self-adjoint extensions, so the formal expression alone does not select a physical Hamiltonian.
  • No bounded-multiplier repair of the metric can restore coercivity, ruling out a whole class of attempted pseudo-Hermitian fixes.
  • If point spectrum exists in this completion, it must come from eigenvectors other than the original BBM functions $\psi_z$.
  • Any BBM-type construction that hopes to recover zeta zeros as point spectrum must leave the $L^2$-based completion, for example by adopting a non-$L^2$, rigged, distributional, or unbounded-metric state space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication left implicit is that the obstruction is vector-level rather than extension-level: choosing a different self-adjoint extension cannot restore the missing $\psi_z$, so the BBM proposal's Hilbert-space formulation is not simply under-specified.
  • The sandwich no-go suggests that any repair of the BBM similarity must alter the shift factor $\Delta$ itself or the factorization $\Delta^{-1}D\Delta$, not merely replace the metric by $\Delta^\dagger h(D)\Delta$.
  • A testable extension is to ask whether an unbounded metric of the form $\Delta^\dagger h(D)\Delta$, with $h$ unbounded, can define a Hilbert topology that contains the critical-line functions $x^{-1/2-i\gamma}$; the paper neither constructs nor excludes such a space.
  • The deficiency-index phenomenon, with infinitely many extensions, a purely continuous free realization, and point spectrum in the adjoint, is likely generic for formal Hamiltonians built from a non-coercive similarity and could be probed in other shift-difference models.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

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Summary. The paper analyzes the Hilbert-space completion induced by the Bender--Brody--Muller candidate metric \eta_0=\Delta^\dagger\Delta on the standard half-line core C_c^\infty(0,\infty). It shows that \eta_0 is positive with trivial kernel but is not coercive, that the completion is canonically unitarily equivalent to L^2(\mathbb R_+) via \Delta, and that the free transported realization of the BBM formal expression is unitarily equivalent to the dilation generator, with purely absolutely continuous spectrum \mathbb R. It further proves that no bounded sandwich \Delta^\dagger h(D)\Delta is boundedly invertible; that the transported symmetric operator has deficiency indices (\infty,\infty) and its adjoint has every real point as an eigenvalue of infinite multiplicity; and that on the critical line \Delta\psi_z=x^{-z} is not in L^2, so the BBM eigenfunctions do not belong to the completion. The paper concludes that no self-adjoint realization in this L^2-based completion can realize the original BBM eigenfunction/boundary-condition mechanism for Riemann zeros, while explicitly excluding non-L^2, rigged-Hilbert-space, and distributional formulations from its scope.

Significance. If correct, this settles the status of the original BBM proposal within the standard L^2-based quasi-Hermitian framework: the metric topology alone excludes the candidate eigenfunctions, independently of the choice of self-adjoint extension. The paper provides a complete, internally consistent operator-domain analysis with explicit deficiency functions, a Mellin reduction, and a realization-independent obstruction. It also contributes standalone spectral facts, namely the absence of boundedly invertible metric sandwiches and the contrast between the adjoint's real point spectrum and the purely continuous free extension. The main limitation, which is properly acknowledged in the manuscript, is that the negative conclusion is relative to the L^2-based completion and does not classify rigged-Hilbert-space or distributional formulations. The supplemental proofs are detailed and, apart from the local issues noted below, check out.

minor comments (3)
  1. [Supplement, Lemma S6] The proof of Lemma S6 states 'Thus ker \Delta^\dagger={0} and Ran \Delta=L^2(\mathbb R_+)'; the second equality is false, since ker \Delta^\dagger={0} only implies that Ran \Delta is dense, and in fact Ran(I-S) is not all of L^2 (for example, the indicator 1_{(0,1)} is not in the range). The desired conclusion that \Delta D_0 is dense still follows: Ran \Delta is dense, D_0 is dense, and \Delta is bounded, so \Delta D_0 is dense in Ran \Delta and hence in L^2. Please correct this intermediate statement.
  2. [Supplement, inner-product convention] The displayed definition '\langle u,v\rangle=\int u(x)v(x)\,dx, linear in the second argument' is not a positive-definite Hilbert inner product as written; the adjoint computations that follow use the convention \langle u,v\rangle=\int \overline{u(x)}v(x)\,dx (linear in the second argument). Please fix the displayed definition.
  3. [Supplement, Theorem S15] The claim that a smooth function \psi defines an element of \mathcal H_{\eta_0} only if \Delta\psi\in L^2 is correct via the unitary identification of Theorem S7, but it would be helpful to state this explicitly rather than as a pointwise condition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the missing-eigenstate theorem is a direct consequence of the explicitly defined L2-based completion and standard external identities, not an assumed conclusion.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The completion is defined by the norm ||psi||_eta0 = ||Delta psi||_L2 from the candidate metric eta0 = Delta^dagger Delta; the unitary identification of the completion with L2(R+) is proved from the density of Delta C_c^infty(0,infty) (Lemma S6, Theorem S7), using only standard facts about the shift and Paley-Wiener. The free transported realization eH_BBM = Delta^{-1} D Delta is explicitly presented as one choice among many, not as a forced consequence, and the paper proves rather than assumes its spectral properties via Mellin transform. The central absence statement (Theorem S15, Corollary S16) follows from the Hurwitz identity Delta psi_z = x^{-z} for x > 1 and the elementary non-integrability of x^{-1/2}, both external to the paper's own framework. No parameter is fitted and no related quantity is predicted from fitted data. The citations to BBM, Bellissard, Moxley, and others are historical context and do not carry the proof; there are no self-citations by the current author and no imported uniqueness theorem. The de Branges remark is explicitly non-classificatory and therefore not load-bearing. The only fragile point is the explicitly scoped choice to analyze the standard L2-based completion rather than rigged-Hilbert-space, distributional, or unbounded-metric formulations; the paper repeatedly and clearly disclaims those alternatives, so this is a modeling assumption with stated scope, not a circular derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests only on standard operator-theoretic tools plus the explicit choice of the L^2-based completion; no free parameters or new entities are introduced.

assumptions (4)
  • standard math Paley-Wiener theorem identifies L^2(R+) with H^2(C^-) via Fourier transform, and Δ becomes multiplication by 1-e^{-ik}.
    Used to compute the metric symbol (Eq. 8) and the non-coercivity probes (Eqs. 9-10).
  • standard math Hurwitz zeta identity ζ(z,x) - ζ(z,x+1) = x^{-z}.
    Used in Theorem S15 to compute Δψ_z = x^{-z}; a classical identity from zeta-function theory.
  • standard math The operator D = -i(2x∂_x+1) is essentially self-adjoint on C_c^∞(0,∞) and its Mellin transform is multiplication by 2τ.
    Used in Theorem S9 for the spectrum of the free realization.
  • domain assumption The relevant state space for the BBM pseudo-Hermitian construction is the standard L^2-based completion induced by η0.
    The paper restricts to this completion and explicitly excludes rigged-Hilbert-space or distributional formulations; the negative conclusion is relative to this choice.

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Pith. "Pith review of Metric completion of the Bender--Brody--M\"uller Hamiltonian: dilation spectrum and missing eigenstates." pith.science (2026). https://pith.science/paper/NUH663PF

@misc{pith2026260719067,
  author       = {Pith},
  title        = {Pith review of: Metric completion of the Bender--Brody--M\"uller Hamiltonian: dilation spectrum and missing eigenstates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUH663PF}},
  note         = {Machine review of arXiv:2607.19067}
}
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.

Figures

Figures reproduced from arXiv: 2607.19067 by the authors.

Figure 1
Figure 1. FIG. 1. Metric degeneracy of the BBM form [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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