A self-adjoint operator whose eigenvalues are the Riemann zeros
In plain words
If the Riemann zeros were the energy levels of a quantum system, they would all be real numbers, which is the Riemann hypothesis. The task is to find such a system.
Precise statement
Find a self-adjoint operator $H$ on a Hilbert space, defined independently of the zeros, with spectrum $\{\gamma_n\}$ where $\zeta(1/2 + i \gamma_n) = 0$, and prove the spectral identity. The Berry-Keating proposal $H = xp$ with suitable boundary conditions reproduces the smooth counting function $N(E) \sim (E/(2 \pi)) \operatorname{log}(E/(2 \pi e)) + 7/8$ but not the individual zeros. Answer: an explicit operator with proof.
What would settle it
A proof that the spectrum of an explicitly constructed self-adjoint operator equals the set of nontrivial zeros.
Status in the literature
Unverified note
Connes and collaborators proposed trace-formula and prolate-operator constructions (1998 to 2026) whose low spectra approach the zeros numerically; no proof of the identity.