Green's function: Dirac-delta point scatterer where point sorce is located

[vacaloca]

The differential equation is as follows:

[d/dx^2 + k^2 - tau * dirac_delta(x-x') ] * G(x,x') = dirac_delta(x-x')

where tau is a complex valued scattering strength, and assuming scattering waves at infinity. The problem asks to derive the solution to this equation.

I've looked over Green's function theory, and I'm stumped. I don't think I can use a closed form solution or Sturm Liouville theory because of the impact of the delta function.

Would it be related to transforming the dirac delta function? something like...

dirac_delta(x-x') = 1/(sqrt(2*pi)*int( 1/(sqrt(2*pi)*e^[j*Beta*x'])*e^[-j*Beta*x],-inf,inf,Beta)

 

[Thaakisfox]

Try working in Fourier space ;) you will get a nasty integral for the Green's function, but it looks being solvable :)