It is well known that in $k$-space, a Green's function usually has asymptotic behaviour $\frac{1}{\omega}$ as $\omega \to \infty$. Are there any similar result in position space? This idea is inspired by considering the Green's function of free electron gas: $$ G(\mathbf{r}, \mathbf{r^\prime}) = -\frac{m}{2\pi\hbar^2}\frac{e^{ik|\mathbf{r}-\mathbf{r^\prime}|}}{|\mathbf{r}-\mathbf{r^\prime}|}\to\frac{e^{ikr}}{r}e^{-ik\hat{r}r^\prime} $$ when $\mathbf{r}\to\infty$. This result is very useful in deriving the scattering cross section of free electrons. I'm now considering about a similar problem in sold where the Green's function can be written as (Bloch Representation): $$ G(\mathbf{r}, \mathbf{r^\prime})=\int\mathrm{d}\mathbf{k}\frac{\psi_\mathbf{k}(\mathbf{r})\psi^*_\mathbf{k}(\mathbf{r^{\prime}})}{\epsilon-\epsilon_\mathbf{k}+i0^+} $$
I think it may be useful to do some asymptotic analysis on this Green's function.
