Hamiltonian with Energy Levels Corresponding to Riemann Zeros

Abstract

A Hamiltonian with eigenenergy \( E_n = ρ_n(1 - ρ_n) \) has been constructed, where \( ρ_n \) denotes the \( n \)-th non-trivial zero of the Riemann zeta function. To construct such a Hamiltonian, we generalize the Berry-Keating paradigm and encode number-theoretic information into the Hamiltonian using modular forms.Although our construction does not resolve the Hilbert-Pólya conjecture (since the eigenstates corresponding to \( E_n \) are \emph{not} normalizable), it provides a novel physical perspective on the Riemann Hypothesis (RH). In particular, we propose a physical interpretation of RH, which could offer a potential pathway toward its proof.

Figures (2)

• Figure 1: The double well structure of the geometric potential $V_{ R}(x, y)$. It can be clearly seen that there are two potential wells located at the origin and $(x, y) = (1, 0)$. Note here to demonstrate the structure of $V_{ R}$ we actually plot the logarithm of it.
• Figure 2: The probability density of the first two eigenstates, corresponding to the first two Riemann zeros: (a) $\rho_1 = 0.5 + \mathbbm{i} 14.13472\ldots$ and (b) $\rho_2 = 0.5 + \mathbbm{i} 21.02203\ldots$, respectively, is shown. At the zeros of $V_{ R}$, namely $z = 0$ and $z = 1$ (where $z := x + \mathbbm{i} y$), the wave function indeed vanishes. We plot the logarithm of the probability density so that the nodal line, on which the wave function $\psi(z)$ vanishes, is clearly visible.