Hamiltonian for the Zeros of the Riemann Zeta Function

Abstract

A Hamiltonian operator H is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of H is 2xp, which is consistent with the Berry-Keating conjecture. While H is not Hermitian in the conventional sense, iH is PT symmetric with a broken PT symmetry, thus allowing for the possibility that all eigenvalues of H are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that H is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.