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Prof. Philip Mannheim, Department of Physics, University of ConnecticutTime advance and probability conservation in PT-symmetric quantum mechanics While time delay is associated with the decay of states with energy eigenvalue \(E_0-i\Gamma\), probability conservation is not lost if one also includes the excitation of the decay products, states that have complex conjugate energy eigenvalue \(E_0+i\Gamma\). In consequence, because of probability conservation the standard time delay associated with decay must be accompanied by an equal and opposite time advance (negative time delay) for excitation. Thus when a photon excites an atom the spontaneous emission of a photon from the excited state must occur without any time delay at all. An effect of this form together with an associated negative time delay have recently been reported by Sinclair et al., PRX Quantum 3, 010314 (2022) and Angulo et al., arXiv:2409.03680 [quant-ph]. The presence of states with complex energy entails lack of Hermiticity, while the presence of complex conjugate pairs of energy eigenstates entails the presence of antilinear \(PT\) symmetry (\(P\) is parity, \(T\) is time reversal). Our analysis shows that the nonrelativistic square well with a real potential is not a Hermitian system but instead is a \(PT\) symmetric system, possessing \(PT\) symmetry in both the bound and scattering sectors. In the scattering sector there are complex conjugate pairs of energy eigenvalues. Despite this there is just the same number of observable resonances as in the familiar Breit-Wigner description of decays. P. D. Mannheim: Time advance and probability conservation in PT-symmetric quantum mechanics, Phys. Rev. D Lett., 112, L031903 (2025), https://arxiv.org/abs/2504.12068 P. D. Mannheim: PT symmetry and the square well potential: Antilinear symmetry rather than Hermiticity in scattering processes, https://arxiv.org/abs/2505.07798