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, i ^H 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 de ne 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.