Critique of Breit-Wigner resonance scattering

In the standard Breit-Wigner approach to scattering the phase shift is to have a form $\tan\delta_{\rm BW} =\Gamma_1/(E_1-E)$ at a real energy resonance. This leads to complex energy poles in the scattering amplitude at $E_{\rm BW}=E_1-i\Gamma_1$, poles that are identified with unstable physical particles. By solving the square well scattering problem we identify some challenges to this approach. We find that setting $\tan\delta_{\rm BW} =\Gamma_1/(E_1-E)$ is not always a good description of the real energy scattering amplitude, that $\Gamma_1$ can be negative, that $E_{\rm BW}$ is not in fact an energy eigenvalue (and thus not a physical particle), and that states that decay in energy possess spatial wave functions that unacceptably grow exponentially. All of this is resolved by noting that because of its antilinear $PT$ symmetry solutions to the square well Schr\"odinger equation appear in complex conjugate energy pairs $E_{\mp}=E_2\mp i \Gamma_2$ with $E_- \neq E_{\rm BW}$, doing so in a way that gives a time independent probability amplitude that neither grows nor decays in time or space, and leads to just one now observable physical resonance not two.