Formally Self-Adjoint Hamiltonian for the Hilbert-P\'olya Conjecture

TL;DR

This paper proposes a novel self-adjoint Hamiltonian model that links its eigenvalues to the nontrivial zeros of the Riemann zeta function, aiming to provide a new perspective on the Riemann hypothesis.

Contribution

It introduces a two-dimensional Hamiltonian coupling the Berry-Keating operator with the number operator, using a unitary transformation to relate to the zeta zeros.

Findings

Constructed a Hamiltonian with eigenvalues at zeta zeros

Demonstrated boundary conditions relate to zeta function zeros

Proposed a pathway to prove the Riemann hypothesis

Abstract

We construct a formally self-adjoint Hamiltonian whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. We consider a two-dimensional Hamiltonian which couples the Berry-Keating Hamiltonian to the number operator on the half-line via a unitary transformation. We demonstrate that the unitary operator, which is composed of squeeze (dilation) operators and an exponential of the number operator, confines the eigenfunction of the Hamiltonian to one dimension as the squeezing parameter tends towards infinity. The Riemann zeta function appears at the boundary of the resulting confined wave function and vanishes as a result of the imposed boundary condition. If the formal argument presented here can be made more rigorous, particularly if it can be shown rigorously that the Hamiltonian remains self-adjoint under the imposed boundary condition, then our approach has the potential to imply that the Riemann hypothesis is true.