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mp.mathematical physics – Formula from a figure having one dirac delta to a figure having two dirac delta

I’d love to question at which situation I can rework a method with one delta duty to a figure with two delta duty.

Suppose a somatic system has a quasi-continuous energy-level spectrum $E_1$, $E_2$, $ldots$, $E_infty$, the response $sigma$ of the system is expressed as:

$sigma=sum_{mn}|p_{mn}|^2delta(E_m-E_n)(frac{partial f}{partial E})_{E_n}$.

the place $|p_{mn}|^2$ denotes transtion likelihood, and $f$ is a fermi-Dirac duty. This could breathe written as:

$sigma=int_{-infty}^{infty}dEsum_{mn}|t_{mn}|^2delta(E-E_n)delta(E-E_m)frac{partial f}{partial E}$.

Is the insertion of an addition delta duty rectify?

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