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Always remember & never forget . . .
... the Fermi Golden Rule:
$$B(\omega,v-v_*)=\frac{1}{8\pi^2}|\omega\cdot(v-v_*)||{\widehat \phi}
((\omega\cdot(v-v_*))\omega)|^2$$
---you know, where the hat $\sphat$ denotes the Fourier transform and $\phi$ is a two-body interaction potential.
$$B(\omega,v-v_*)=\frac{1}{8\pi^2}|\omega\cdot(v-v_*)||{\widehat \phi}
((\omega\cdot(v-v_*))\omega)|^2$$
---you know, where the hat $\sphat$ denotes the Fourier transform and $\phi$ is a two-body interaction potential.
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Of course, all the other stuff in your post was complete gibberish to me.
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On the other hand, it looks good.
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