Self-adjointness of the Bender–Brody–Müller operator

Establish whether the Hamiltonian proposed by Carl M. Bender, Dorje C. Brody, and Markus P. Müller (Physical Review Letters, 118 (13) 130201, 2017), whose eigenvalues match the nontrivial zeros of the Riemann zeta function, is self-adjoint on an appropriate domain in a Hilbert space.

Background

The paper reviews operator-based approaches to the Riemann Hypothesis and highlights the Bender–Brody–Müller construction, which proposes a Hamiltonian intended to reproduce the nontrivial Riemann zeros as eigenvalues. For the Hilbert–Pólya strategy to apply rigorously, the relevant operator must be self-adjoint.

Despite numerical and heuristic evidence that the spectrum aligns with the Riemann zeros, the self-adjointness of the proposed operator has not yet been rigorously established, leaving a critical gap in the approach.