Time advance and probability conservation in PT -symmetric quantum mechanics

When excited states decay with energy E₀-iΓ the time evolution operator U(t)=e^-iHt does not obey U^(t)U(t)=I. Nonetheless, probability conservation is not lost if one includes both decay and excitation with energy E₀+iΓ, though it takes a different form. Specifically, if the eigenspectrum of a Hamiltonian is complete, then due to $CPT$ symmetry, a symmetry that holds for all physical systems, there must exist an operator V that effects VHV^-1=H^, so that V^-1U^(t)VU(t)=I. In consequence, because of probability conservation the time delay associated with decay must be accompanied by an equal and opposite time advance for excitation. Thus when a photon excites an atom the spontaneous emission of a photon from the excited state must occur without any decay time delay at all. An effect of this form together with an associated negative time delay has recently been reported by Sinclair et al. [PRX Quantum 3, 010314 (2022)] and Angulo et al. [arXiv:2409.03680].