Time advance and probability conservation in PT-symmetric quantum mechanics and the square well potential

When excited states decay with energy $E_0-i\Gamma$, the time evolution operator $U(t)=e^{-iHt}$ does not obey $U^{\dagger}(t)U(t)=I$. Nonetheless, probability conservation is not lost if one includes both decay and excitation with energy $E_0+i\Gamma$, 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 $VHV^{-1}=H^{\dagger}$, so that $V^{-1}U^{\dagger}(t)VU(t)=I$. As a 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, have recently been reported by Sinclair et al., PRX Quantum 3, 010314 (2022), and Angulo et al., arXiv:2409.03680 [quant-ph]. We show that the non-relativistic square well problem with a real potential possesses $PT$ symmetry in both the bound and scattering sectors, with complex conjugate pairs of energy eigenvalues in the scattering sector. In addition, we show that the square well scattering threshold branch point is an exceptional point (a characteristic of systems with an antilinear symmetry such as $PT$), a point at which the Hamiltonian becomes of non-diagonalizable, and thus manifestly non-Hermitian, Jordan-block form. The square well potential, one of the oldest known quantum-mechanical systems, provides an explicit realization of how antlinearity is more general than Hermiticity.

References:

Phys. Rev. D 112, L031903 (2025). (arXiv:2504.12068 [quant-ph]).

ArXiv: 2505.07798 {quant-ph].