Density-density interaction in Luttinger liquids

[AdeBlackRune]

Hi, I'm reading ad article of Lal et al (junction of several weakly interaction quantum wires: a renormalization group study). It's quite cleare but there are some row I'm not good to understand. Could someone help me to understand how obtain equation (6) from equation (4)? I don't know how to obtain from a double integral in x and y a single integral in x under the assumption of a finite range potential.

 

[OhYoungLions]

Looks like you could solve it by Fourier transform, and then take only the Fourier components of the potential V(0) and V(2kf) to be nonzero. Then transform it back.

 

[shawl]

Agree with OhYoungLions.

Fourier transformation.

The variable x in Eq. (6) strictly accords with momentum space。