Resonances/decaying states and the mathematics of quantum physics

There is sufficient experimental evidence that a Breit–Wigner scattering resonance of width ΓΓ is the same physical entity as an exponentially decaying Gamow state of lifetime τ=ℏ/Γ.τ=ℏ/Γ. In order to derive a Gamow ket with exponential time evolution from the Breit–Wigner scattering amplitude of the S-matrix pole, one has to make assumptions about the mathematical properties of the energy wave function for the prepared in-state φ+φ+ and the detected out-“state” ψ−ψ− of a resonance scattering experiment. These mathematical properties identify the space of in-state energy-wave functions as {φ+(E)}=H2¯ and of out-state wave functions as {ψ−(E)}=H+2 as the Hardy function spaces of the lower and upper complex energy plane. The semigroup-time asymmetry t0=0