Averaged t-matrix Approximation
D ( E ) − D 0 ( E ) = − 2 π I m T r [ x τ A N ⟨ τ ⟩ ∑ q T q A T d τ A − 1 d E + y τ B N ⟨ τ ⟩ ∑ q T q A T d τ B − 1 d E − 1 N ∑ q T q A T d B q d E ] = − 2 π I m T r [ ⟨ τ ⟩ − 1 { x τ A d τ A − 1 d E + y τ B d τ B − 1 d E } ∑ q T q A T − ∑ q T q A T d B q d E ] = 2 π I m T r ∑ q T q A T [ ⟨ τ ⟩ − 1 { x τ A d τ A − 1 d E + y τ B d τ B − 1 d E } + d B q d E ] {\displaystyle {\begin{matrix}&D(E)-D_{0}(E)&\\\ &=&-{2 \over {\pi }}\mathrm {ImTr} \left[x{\tau _{A} \over {N{\left\langle \tau \right\rangle }}}\sum _{\mathbf {q} }T_{\mathbf {q} }^{AT}{d\tau _{A}^{-1} \over {dE}}+y{\tau _{B} \over {N{\left\langle \tau \right\rangle }}}\sum _{\mathbf {q} }T_{\mathbf {q} }^{AT}{d\tau _{B}^{-1} \over {dE}}-{1 \over N}\sum _{\mathbf {q} }T_{\mathbf {q} }^{AT}{dB_{\mathbf {q} } \over {dE}}\right]\\\ &=&-{2 \over {\pi }}\mathrm {ImTr} \left[{\left\langle \tau \right\rangle }^{-1}\left\{x\tau _{A}{d\tau _{A}^{-1} \over {dE}}+y\tau _{B}{d\tau _{B}^{-1} \over {dE}}\right\}\sum _{\mathbf {q} }T_{\mathbf {q} }^{AT}-\sum _{\mathbf {q} }T_{\mathbf {q} }^{AT}{dB_{\mathbf {q} } \over {dE}}\right]\\\ &=&{2 \over {\pi }}\mathrm {ImTr} \sum _{\mathbf {q} }T_{\mathbf {q} }^{AT}\left[{\left\langle \tau \right\rangle }^{-1}\left\{x\tau _{A}{d\tau _{A}^{-1} \over {dE}}+y\tau _{B}{d\tau _{B}^{-1} \over {dE}}\right\}+{dB_{\mathbf {q} } \over {dE}}\right]\end{matrix}}}
となる。