Hamiltonian for the Zeros of the Riemann Zeta Function

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 2 x p , which is consistent with the Berry-Keating conjecture. While H ^ is not Hermitian in the conventional sense, i H ^ is P T symmetric with a broken P T 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.