Metric completion of the Bender--Brody--Müller Hamiltonian: dilation spectrum and missing eigenstates

Technical Summary: Metric Completion of the Bender–Brody–Müller Hamiltonian

Problem Statement
The paper addresses the structural validity of the Bender–Brody–Müller (BBM) proposal for the Hilbert–Pólya conjecture. The BBM Hamiltonian, $H_{\text{BBM}} = \Delta^{-1} D \Delta$, is a non-Hermitian operator intended to have a real spectrum proportional to the imaginary parts of the non-trivial zeros of the Riemann zeta function. Here, $D = \hat{x}\hat{p} + \hat{p}\hat{x}$ is the dilation generator, and $\Delta = 1 - \hat{S}$ is a shift-difference operator. The proposal relies on a pseudo-Hermitian framework where a candidate metric operator $\hat{\eta} = \sin^2(\hat{p}/2) = \Delta^\dagger \Delta / 4$ renders the Hamiltonian self-adjoint.

The central problem is that while $\hat{\eta}$ is positive and bounded, its inverse is unbounded. Previous works (e.g., Bellissard, Moxley) noted that the BBM eigenfunctions are not $L^2$ states on the critical line and that the metric topology might be ill-defined. This paper rigorously constructs the Hilbert space completion induced by the BBM metric on the standard dense core $C_c^\infty(0, \infty) \subset L^2(\mathbb{R}^+)$ to determine:

1. The topology of the physical state space.
2. The spectral properties of the resulting self-adjoint realizations.
3. Whether the original BBM candidate eigenfunctions (related to Hurwitz zeta functions) exist within this completed space.

Methodology
The analysis proceeds through the following steps:

1. Metric Formulation: The study focuses on the quadratic form $\eta_0 = \Delta^\dagger \Delta$ defined on the test domain $D_0 = C_c^\infty(0, \infty)$. The norm is defined as $\|\psi\|_{\eta_0} = \|\Delta \psi\|_{L^2}$.
2. Coercivity Analysis: Using the Fourier transform and the Paley–Wiener theorem, the paper analyzes the symbol of the metric, $\eta_0(k) = |1 - e^{-ik}|^2 = 2(1 - \cos k)$. It demonstrates that while $\eta_0$ has a trivial kernel, it is not coercive (bounded below) because the symbol vanishes at $k \in 2\pi\mathbb{Z}$.
3. Hilbert Space Completion: The space $D_0$ is completed under the $\|\cdot\|_{\eta_0}$ norm. The map $\Delta: D_0 \to L^2(\mathbb{R}^+)$ is shown to be an isometry with a dense image, establishing a canonical unitary equivalence between the completed space $\tilde{\mathcal{H}}_{\eta_0}$ and $L^2(\mathbb{R}^+)$.
4. Operator Realization: The formal Hamiltonian is realized in this completed space. The "free transported" realization is defined as $\tilde{H}_{\text{BBM}} = \Delta^{-1} D \Delta$, where the domain is transported via the unitary map $\Delta$.
5. Spectral Analysis: The spectrum of the free realization is determined via the Mellin transform. The paper also investigates the deficiency indices of the symmetric core $T = D|_{\Delta D_0}$ and the spectral properties of its adjoint.
6. Eigenfunction Verification: The paper checks if the proposed BBM eigenfunctions $\psi_z(x) = -\zeta(z, x+1)$ (where $z = 1/2 + i\gamma$) belong to the completed space by examining their shift differences $\Delta \psi_z$.

Key Contributions and Results

• Metric Degeneracy and Non-Coercivity: The paper proves that the BBM metric is not coercive. Specifically, there exist normalized $L^2$ states (constructed via Cauchy kernels concentrated at the zeros of the metric symbol) whose metric length approaches zero. This confirms that the metric topology is strictly weaker than the ambient $L^2$ topology.
• Universal Obstruction: A general result is established for the class of "bounded sandwiches" $\Delta^\dagger h(D) \Delta$. For any bounded measurable function $h$, the operator is not boundedly invertible. This is due to a spectral mismatch: the degeneracy of $\Delta$ is localized in momentum space ($k \in 2\pi\mathbb{Z}$), while $h(D)$ acts in the scale (dilation) variable. No bounded function of the dilation generator can cancel the momentum zeros of the shift operator pointwise.
• Spectral Properties of the Free Realization: The free transported realization $\tilde{H}_{\text{BBM}}$ is unitarily equivalent to the dilation generator $D$. Consequently, its spectrum is purely absolutely continuous, equal to $\mathbb{R}$, with multiplicity one. There are no isolated eigenvalues.
• Deficiency Indices and Adjoint Spectrum: The symmetric operator $T = D|_{\Delta D_0}$ has deficiency indices $(\infty, \infty)$. While its free extension is purely continuous, its adjoint $T^*$ has a point spectrum consisting of the entire real line, where every real number is an eigenvalue of infinite multiplicity.
• Absence of BBM Eigenstates: The most critical result concerns the candidate eigenfunctions. For $z$ on the critical line ($\text{Re } z = 1/2$), the shift difference is $\Delta \psi_z = x^{-z}$. Since $|x^{-z}| = x^{-1/2}$, this function is not square-integrable on $\mathbb{R}^+$. Therefore, $\psi_z \notin \tilde{\mathcal{H}}_{\eta_0}$. The original BBM mechanism, which relies on these specific functions and a boundary condition $\psi_z(0)=0$, fails to produce Riemann-zero eigenstates in this $L^2$-based metric completion.

Significance and Claims
The paper claims to resolve the structural questions surrounding the BBM proposal within the standard framework of bounded metrics and $L^2$-based completions. Its primary conclusion is negative regarding the original BBM mechanism:

• The metric completion fixes the topology of the state space to be unitarily equivalent to $L^2(\mathbb{R}^+)$ via the coordinate $\Phi = \Delta \psi$.
• In this space, the original BBM candidate eigenfunctions do not exist as Hilbert space vectors.
• Consequently, no self-adjoint realization within this completed space can utilize the BBM functions to generate the Riemann zeros as a point spectrum.

The authors emphasize that this result does not rule out all Hilbert–Pólya constructions inspired by BBM, but it specifically rules out the "rescue" of the original proposal simply by choosing a different self-adjoint extension of the operator. To succeed, a surviving route would require abandoning at least one hypothesis of the bounded $L^2$-based construction studied here (e.g., by using a rigged Hilbert space, an unbounded metric, or a distributional state space). The paper also highlights a broader spectral phenomenon: a symmetric operator can possess a purely continuous free extension while its adjoint exhibits a dense point spectrum, a feature independent of the Riemann hypothesis context.