Coulomb term in Potential of ions in a Penning Trap,Green's Function

[aim1732]

How do you write out the potential of an ion in a Penning trap? Specifically, how does the inter-ionic Coulomb interaction come to be written as a Green's function? I found it here : http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.71.87

They do say something about interactions with image charges in the electrodes, but I can not see why a simple expression of all Coulomb pairs is not good enough. They seem to use it well enough in some of the other papers I read.

 

[rigetFrog]

Green's function in E&M just means the the integral k rho/|r-r'| over r' where rho is the change density.

A greens method is a nifty way of solving differential equations by convolving something with the green function. The Green function typically have the form f(r-r').