Can I exchange these operations: Absolute square, limit and time derivative in my specific example involving a Dirac sequence. If yes: why?

Question

I am rederiving some physics stuff (related to Fermi's Golden rule) so I know what the result should be. ( I am a physicist, so I lack some math training.)

However to get this result I have to exchange several operations at some point and I am really unsure why I should be able to do this. The term I have is:

$$ \frac{d}{dt} \left|\lim_{\epsilon \rightarrow 0} \frac{e^{\epsilon t} e^{\mathrm{i} (x -y) t}}{\epsilon + \mathrm{i}( x - y)} \right|^2 $$

If I could just reorder all the operations as much as I want, I would choose: First absolute square, then time derivative, then the limit. This would give me:

$$ \lim_{\epsilon \rightarrow 0} \frac{2 \epsilon e^{2 \epsilon t}}{\epsilon^2 + (x - y)^2} = 2 \pi \delta(x-y) $$ which is what I expected to find at some point in my derivation. But why should this be allowed? Is it allowed in this case?

I tried to read up on when I can exchange two limits when expressing $\frac{d}{dt}$ as a limit. However I think the problem is more special because of the Dirac distribution (?).

I would be happy if someone can give advise here. Thank you very much in advance.

Answer 1

We have

$$\lim_{\epsilon \rightarrow 0} \frac{e^{\epsilon t} e^{\mathrm{i} (x -y) t}}{\epsilon + \mathrm{i}( x - y)}= \frac{e^{i(x-y)t}}{i(x-y)}.$$

Then

$$\left|\lim_{\epsilon \rightarrow 0} \frac{e^{\epsilon t} e^{\mathrm{i} (x -y) t}}{\epsilon + \mathrm{i}( x - y)} \right|^2=\frac{1}{(x-y)^2}.$$

Answer 2

So I could delete my question, because the solution was that my Ansatz was wrong. However I will post my solution instead, just in case other physicist try a similar derivation as mine.

The limit of $\epsilon \rightarrow 0 $ was physically motivated in my case (and introduced into an interaction Hamiltonian). By doing so I should have realised/specified from the beginning that the limit should be only done in the end of my calculation. Ergo after doing the time derivative.