I know $K(a,b,t)$ is the probability amplitude of find a particle that starts at point a in b in a time t later. There is also an expression that sometimes is called green function:
or Fourier transform of Feynman propagator
See: Grosche. Handbook of Feynman path integrals page 149. Keller. the feynman integral page 461.
I want to know if $G(a,b,E)$ could be the amplitude that a particle of energy $E$ at the initial point $a$ will appear at some (arbitrary) time at $b$. It seems that Martin Schaden and Larry Spruch use this interpretation in http://arxiv.org/abs/cond-mat/0304290v1 but I have not found this in any book of quantum mechanics.