I'm a little confused about the interpretation of the Greens function. My understanding is that the Greens function is related to the solution of differential equations, so it should be unique.
For a linear second order homogeneous ODE with separated boundary conditions we can write the Greens function as $$G(x,x_0)= \begin{cases} \frac{u_1(x)u_2(x_o)}{p_2(x_0)W(x_0)},\quad x<x_0 \\ \frac{u_1(x_0)u_2(x)}{p_2(x_0)W(x_0)}, \quad x>x_0 \end{cases}$$ where $u_1$ and $u_2$ are solutions of the ODE satisfying the left and right boundary conditions respectively, $p_2(x)$ is the polynomial multiplying the second derivative term and $W(x)$ is the Wronskian.
Here's where I'm getting confused. For linear homogeneous ODEs the Wronskian isn't unique, in this case it's given by $$W(x) = C\exp\left[-\int^x p_1(s) ds\right]$$ where $p_1(x)$ is the polynomial multiplying the first derivative term and $C$ is an arbitrary integration constant.
Doesn't this mean the Greens function isn't unique (since it depends on $W(x)$)?
The Wronskian $$W=u_1u_2'-u_1'u_2$$ is calculated with respect to the same solutions $u_1,u_2$ that are used to define the Green function. This corresponds to a specific value of the constant $C$ and ensures uniqueness.