(a) Solve the generalized \(K\)-matrix equation [i.e., (10.21) with
\(G_{0}^{+}\)replaced by \(\left.G_{0}\right]\) for \(K\) when \(V_{D-p}\), is
approximated by a factorizable potential, i.e., \(V_{p-p} \approx \approx\) \(A
u_{p} u_{p^{*}}\). (Hint: Try guessing the form of \(K .\) If you get stuck, go
back to (10.19) and carry out the sum directly.)
(b) In the integral which occurs in the solution for \(K\) in \((a)\) - call this
integral \(I(\mathrm{q}, \omega)\) - carry out the integration over frequency
\(\epsilon^{\prime \prime} .\) Show that \(I\) is the sum of two contributions:
\(1^{+}\)coming from \(G_{0}^{+}\)(i.e., particles) and \(I^{-}\)coming from
\(G_{0}^{-}\) (i.e., holes).
(c) Evaluate \(I^{+}\)and \(I^{-}\)in \((b)\) in the case \(\omega=0, q=0\). Take
\(u_{p}=1\) for \(0<|p|k_{F}\right)\) and \(u_{p}=0\) for \(|p|>w .\) Use
theorem (3.76)
(d) Show that in the case of low density, \(I^{+}>I^{-}\), i.e., contribution
from particle lines much greater than that from hole lines.
