What is the meaning of the Fourier transform of Feynman propagator?

I know $K(a,b,t)$ is the probability amplitude that a particle that starts at point $a$ is found at point $b$ at a time $t$ later. There is also an expression that sometimes is called green function:
$$G(a,b,E)=(i/\hbar)\int_{-\infty}^\infty\;\exp(iEt/\hbar)\;K(a,b,t)\;dt$$

or Fourier transform of Feynman propagator
See: Grosche. Handbook of Feynman path integrals page 149. Keller. the feynman integral page 461.
http://arxiv.org/abs/cond-mat/0304290v1

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.

In the usual text books, the Fourier transform is taken over position and time $(\vec{x},t)$ to get energy and momentum $(E,\vec{p})$. This is done by four integrations, one each getting rid of one of the four variables $x_1, x_2, x_3, t$, and replacing with the corresponding one of the four Fourier transformed variables $p_1, p_2, p_3, E$. Mathematically, nothing is wrong with only doing one of the four transforms, and the interpretation is clear.
• What is the physical interpretation of the propagator as a function of $E$ and $p$? – ChickenGod Apr 15 '13 at 1:34