ABSTRACT
All our Green’s function approximations so far have been based on the idea that the potential is small. Even the shielded potential approximation depends on there being a dimensionless parameter proportional to the strength of the interaction, which is small. For zero-temperature fermions, this parameter is r s = 1 a 0 3 4 π n 1 / 3 , $ r_{s} = \frac{1}{{a_{0} }}\left( {\frac{3}{4\pi n}} \right)^{1/3} , $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math14_1.tif"/> and in the classical limit, it is 1 r D 3 4 π n 1 / 3 · $ \frac{1}{{r_{\text{D}} }}\left( {\frac{3}{4\pi n}} \right)^{1/3} \cdot $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math14_2.tif"/> However, in many situations of practical interest, the potential is not small but nonetheless the effects of the potential are small because the potential is very short-ranged. For example, a gas composed of hard spheres with radius r 0 has the potential v ( r ) = 0 for r > r 0 1 for r < r 0 $$ v(r) = \left\{ {\begin{array}{*{20}l} 0 \hfill & {{\text{for}}\,\,r~> r_{0} } \hfill \\ 1 \hfill & {{\text{for}}\,\,r~ < r_{0} } \hfill \\ \end{array} } \right. $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/math14_1.tif"/>
