AI 中文总结
研究BBM哈密顿量的度量完备化，分析其候选度量诱导的希尔伯特完备化及相关谱性质，得出如无界夹层可逆性、算子亏指数等结论，指出原始BBM机制在该完备化中无法产生点谱黎曼零点态。
AI 中文摘要
本德尔 - 布罗迪 - 米勒（BBM）哈密顿量被提议作为一个非厄米特希尔伯特 - 波利亚算子。我们分析了由BBM的候选度量\(\hat\eta=\sin^2(\hat p/2)=\Delta^\dagger\Delta/4\)在标准半直线\(L^2\)核上诱导的希尔伯特完备化。形式\(\eta_0=\Delta^\dagger\Delta\)是正定的且核平凡，但不具有强制性。在范数\(\|\psi\|_{\eta_0}=\|\Delta\psi\|\)下完备化\(C_c^\infty(0,\infty)\)得到一个与\(L^2(\mathbb R_+)\)典范酉等价的希尔伯特空间。其自由自伴实现是伸缩生成元，具有简单的纯绝对连续谱\(\mathbb R\)。分析得出了两个超出BBM问题的谱陈述。首先，没有有界夹层\(\Delta^\dagger h(D)\Delta\)是有界可逆的。其次，传输的对称算子具有亏指数\((\infty,\infty)\)，其伴随算子以每个实点为无限重本征值，而其自由扩展是纯连续的。与实现无关的BBM结论涉及候选本征函数：\(\Delta\psi_z=x^{-z}\)，所以对于\(\operatorname{Re}z = 1/2\)，它们不属于完备空间。因此，在这个基于\(L^2\)的度量完备化中，原始的BBM边界条件/本征函数机制不能产生点谱黎曼零点态。
英文摘要
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 $\hatη=\sin^2(\hat p/2)=Δ^\daggerΔ/4$. The form $η_0=Δ^\daggerΔ$ is positive with trivial kernel but is not coercive. Completing $C_c^\infty(0,\infty)$ in the norm $\|ψ\|_{η_0}=\|Δψ\|$ 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 $Δ^\dagger h(D)Δ$ 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: $Δψ_z=x^{-z}$, 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.
Comments5 pages, 1 figure; Supplemental Material: 6 pages
