In real space and in equilibrium, the retarded Green function $G_{i,j}(t)$ gives the amplitude probability for propagation of a particle transfering from j to i during time t. The Laplace tranformation of this Green function gives $G_{i,j}(z)$ where $z=\omega+i\delta$. I would like to know the meaning of $G_{i,j}(z)$. Can we similarly interpret it as the probability amplitude for a particle to transfer from j to i with energy change as much as $z$?

In real space and in equilibrium, the retarded Green function $G_{i,j}(t)$ gives the amplitude probability for propagation of a particle transfering from j to i during time t. The Laplace tranformation of this Green function gives $G_{i,j}(z)$ where $z=\omega+i\delta$. I would like to know the meaning of $G_{i,j}(z)$. Can we similarly interpret it as the probability amplitude for a particle to transfer from $j$ to $i$ with energy change as much as $z$?