Define $G_n (\mathbf{x},\mathbf{x}')$ as

$$ G_n (\mathbf{x},\mathbf{x}') = \lim_{\epsilon \to 0^{+}} \left[\dfrac{1}{(2 \pi \hbar)^{n}} \int_{\mathbb{R}^{n}} \mathrm{d}^{n}\mathbf{p} \dfrac{e^{i \mathbf{p} \cdot (\mathbf{x}-\mathbf{x'}) / \hbar}}{p^{2}-\hbar^{2} k^{2} - i \epsilon}\right]. $$

Does there exist any relationship between the expressions for $G_n (\mathbf{x},\mathbf{x}')$ for different positive integer values of $n$? In particular, is it possible to determine $G_n (\mathbf{x},\mathbf{x}')$ given an expression for $G_m (\mathbf{x},\mathbf{x}')$, with $m>n$?

This question is motivated by the derivation of an expression for the Lippmann-Schwinger equation in the position representation. (See pp. 379-382 of 'Modern Quantum Mechanics' by Sakurai for a derivation of the expression for $G_3 (\mathbf{x},\mathbf{x}')$.)


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