Green's Function vs. Fundamental Solution

Question

From the texts I've used, the Green's function is of a problem is $G(x,y)$ such that $LG(x,y) = \delta(x-y)$. The fundamental solution is u(x) such that $Lu(x)=\delta(x)$. They seem to be used for the same purpose (solving inhomogenous odes/pdes), but I'm curious how they are related. For example, there is a general methodology for finding Green's functions for simple types of ODEs (i.e. regular Sturm-Liouville problems), but I'm having trouble finding such a methodology for finding fundamental solutions. Also, are the Green's functions and fundamental solutions directly related in some way? I was talking to a friend and his idea was that the Green's function had the convolution built into it (since you use it as a kernel), whereas the fundamental solution is kind of more basic than that since you convolve it with the inhomogenous part.

Answer 1

The difference shows up in operators which are not invariant under translation. For an invariant operator such as the Laplacian, the solution of $\nabla^{2}L(x)=\delta(x)$ can be translated to give a solution of $\nabla^{2}L_{y}(x)=\delta(x-y)$ by setting $L_{y}(x)=L(x-y)$. But as soon as you add a potential such as $1/|x|$ to $-\nabla^{2}$ (such as for an atomic Hamiltonian) then the fundamental solution does not generate the Green function. Closed-form Green functions are not going to be so easy to find in general, even though asymptotics may be realistic.