A Hamiltonian for the Hilbert-P\'olya Conjecture

Abstract

We construct a similarity transformation of the Berry-Keating Hamiltonian, whose eigenfunctions vanish at the Dirichlet boundary as a consequence of the Riemann hypothesis (RH) so that the eigenvalues correspond to the imaginary parts of the nontrivial zeros of the Riemann zeta function. Conversely, if one is able to prove the reality of the eigenvalues, which corresponds to proving that the similarity transformation is bounded and boundedly invertible on the domain where the Berry-Keating Hamiltonian is self-adjoint, then the RH follows. In an attempt to show the latter heuristically, we first introduce an $su(1,1)$ algebra and then define an effective Hamiltonian in the Mellin space, where the Dirichlet boundary condition manifests itself as an integral boundary condition. The effective Hamiltonian can be transformed into the Berry-Keating Hamiltonian, $\hat{H}_\text{BK}$, without altering the domain on which $\hat{H}_\text{BK}$ is self-adjoint. In an essence, the nontrivial zeros of the Riemann zeta function follow from the eigenvalue equation, $\hat{H}_\text{BK} \, h_s (z) = \varepsilon_s \, h_s (z)$, with the integral boundary condition $\int_0^\infty dz \, (1+ e^z)^{-1} h_s(z) = 0$.Comment: 4 pages. arXiv admin note: substantial text overlap with arXiv:2211.0189