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

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 $\hat{\hat{I}}\cdot ={\sin }^2(\hat{p}/2)=\hat{I}{”}^{†}\hat{I}”/4$. The form $\hat{I}{\cdot }_0=\hat{I}{”}^{†}\hat{I}”$ is positive with trivial kernel but is not coercive. Completing $C_c^{\infty }(0,\infty )$ in the norm $\Vert \ddot{I}ˆ{\Vert }_{\hat{I}{\cdot }_0}=\Vert \hat{I}”\ddot{I}ˆ\Vert$ 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 $\hat{I}{”}^{†}h(D)\hat{I}”$ 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: $\hat{I}”\ddot{I}{ˆ}_z=x^{-z}$, so for $\mathrm{Re}z=1/2$ 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.

5 pages, 1 figure; Supplemental Material: 6 pages