Fourier Transform of a productof Green functions

[Physicslad78]

Guys, how do u get the Fourier transform of a product of Greens Functions?I have to get Fourier transform of:

G$$_{el}$$(k+q,$$\tau$$-$$\tau1$$)*G$$_{el}$$(k,$$\tau1$$) where $$\tau$$ and $$\tau1$$ are two different times ($$\tau$$>$$\tau1$$) and q is phonon momentum and k is electron momentum...


Thanks

 

[SpectraCat]

*cough* homework problem *cough*

...

*cough* convolution theorem *cough*

 

[Entropia]

for your question! To calculate the Fourier transform of a product of Green functions, we can use the convolution theorem. This theorem states that the Fourier transform of a product of two functions is equal to the convolution of their individual Fourier transforms. In this case, we can write the product of Green functions as a convolution integral:

G_{el}(k+q,\tau-\tau1)*G_{el}(k,\tau1) = \int_{-\infty}^{\infty} G_{el}(k+q,\tau-\tau') * G_{el}(k,\tau') d\tau'

where \tau' is a dummy variable. Now, we can take the Fourier transform of both sides of this equation:

\mathcal{F}[G_{el}(k+q,\tau-\tau1)*G_{el}(k,\tau1)] = \mathcal{F} \left[ \int_{-\infty}^{\infty} G_{el}(k+q,\tau-\tau') * G_{el}(k,\tau') d\tau' \right]

Using the convolution theorem, the right side of the equation becomes:

\mathcal{F}[G_{el}(k+q,\tau-\tau1)*G_{el}(k,\tau1)] = \mathcal{F}[G_{el}(k+q,\tau-\tau')] * \mathcal{F}[G_{el}(k,\tau')]

Substituting back into the original equation, we get:

\mathcal{F}[G_{el}(k+q,\tau-\tau1)*G_{el}(k,\tau1)] = \int_{-\infty}^{\infty} \mathcal{F}[G_{el}(k+q,\tau-\tau')] * \mathcal{F}[G_{el}(k,\tau')] d\tau'

This integral can be evaluated using the Fourier transform of the Green function, which is typically known for specific systems. I hope this helps with your calculations!