多重散乱とは？ わかりやすく解説

${\displaystyle H=\,H_{0}+\sum _{n}V_{n}}$

${\displaystyle H=[H_{0}+{\tilde {V}}(z)]+\sum _{n}[V_{n}-{\tilde {V}}_{n}(z)]={\tilde {H}}(z)+v(z)}$

${\displaystyle {\tilde {V}}(z)=\sum _{n}{\tilde {V}}_{n}(z)}$

${\displaystyle v(z)=\sum _{n}[V_{n}-{\tilde {V}}_{n}(z)]=\sum _{n}v_{n}(z)}$

${\displaystyle G(z)={\frac {1}{z-{\tilde {H}}-v}}={\frac {1}{(z-{\tilde {H}})\left(1-{\frac {v}{z-{\tilde {H}}}}\right)}}}$

${\displaystyle {\tilde {G}}={\frac {1}{z-{\tilde {H}}}}}$

${\displaystyle G(z)={\tilde {G}}\{1+v{\tilde {G}}+v{\tilde {G}}v{\tilde {G}}+\cdots \}={\tilde {G}}+{\tilde {G}}T{\tilde {G}}}$

${\displaystyle T=v+v{\tilde {G}}v+v{\tilde {G}}v{\tilde {G}}v+\cdots }$

${\displaystyle T=\sum _{n}v_{n}+\sum _{n}v_{n}{\tilde {G}}\sum _{m}v_{m}+\sum _{n}{\tilde {G}}\sum _{m}v_{m}{\tilde {G}}\sum _{p}v_{p}\cdots }$

${\displaystyle t_{n}=v_{n}\{1+{\tilde {G}}v_{n}+{\tilde {G}}v_{n}{\tilde {G}}v_{n}\cdots \}=v_{n}{\frac {1}{1-v_{n}{\tilde {G}}}}=v_{n}[1-v_{n}{\tilde {G}}]^{-1}}$

${\displaystyle t_{n}=v_{n}+v_{n}{\tilde {G}}\{v_{n}+v_{n}{\tilde {G}}v_{n}+v_{n}{\tilde {G}}v_{n}{\tilde {G}}v_{n}\cdots \}=v_{n}+v_{n}{\tilde {G}}t_{n}}$

${\displaystyle T=\,\sum _{n}T_{n}}$

${\displaystyle {\begin{aligned}T_{n}&=v_{n}+v_{n}{\tilde {G}}\sum _{m}v_{m}+v_{n}{\tilde {G}}\sum _{m}v_{m}{\tilde {G}}\sum _{p}v_{p}+\cdots \\&=v_{n}\left\{1+{\tilde {G}}\left[\sum _{m}v_{m}+\sum _{m}v_{m}{\tilde {G}}\sum _{p}v_{p}+\cdots \right]\right\}\\&=v_{n}\left[1+{\tilde {G}}\sum _{m}T_{m}\right]\end{aligned}}}$

${\displaystyle T_{n}=v_{n}+v_{n}{\tilde {G}}T_{n}+v_{n}{\tilde {G}}\sum _{m\neq n}T_{m}=t_{n}\left[1+{\tilde {G}}\sum _{m\neq n}T_{m}\right]}$

${\displaystyle (1-v_{n}{\tilde {G}})T_{n}=v_{n}+v_{n}{\tilde {G}}\sum _{m\neq n}T_{m},\quad t_{n}=v_{n}[1-v_{n}{\tilde {G}}]^{-1}}$

${\displaystyle T=\sum _{n}t_{n}+\sum _{n}t_{n}{\tilde {G}}\sum _{m\neq n}t_{m}+\sum _{n}t_{n}{\tilde {G}}\sum _{m\neq n}t_{m}{\tilde {G}}\sum _{p\neq m}t_{p}+\cdots }$

${\displaystyle T=\,\sum _{n,n'}T_{nn'}}$

${\displaystyle T_{nn'}=t_{n}\delta _{nn'}+t_{n}G_{0}\sum _{m\neq n}T_{mn'}}$

${\displaystyle {\begin{aligned}T_{nn'}&=t_{n}\delta _{nn'}+t_{n}G_{0}\sum _{m\neq n}(t_{m}\delta _{mn'}+t_{m}G_{0}\sum _{p\neq m}T_{pn'})\\&=t_{n}\delta _{nn'}+t_{n}G_{0}t_{n'}+t_{n}G_{0}\sum _{m\neq n}t_{m}G_{0}t_{n'}+\cdots \end{aligned}}}$

${\displaystyle T_{n}=\,\sum _{n'}T_{nn'}}$

${\displaystyle T_{nn'}^{LL'}(\kappa )=\tau _{n}^{l}(\kappa )\left[\delta _{nn'}^{LL'}+\sum _{n_{1},L_{1}}B_{nn_{1}}^{LL_{1}}(\kappa )\cdot T_{{n_{1}}n'}^{{L_{1}}L'}(\kappa )\right]}$

${\displaystyle {\begin{aligned}B_{nn_{1}}^{LL_{1}}(\kappa )&=-\left[4\pi i\kappa \sum _{L_{1}}i^{l-l'-l_{1}}C_{LL'L_{1}}Y_{L_{1}}(\mathbf {R} _{n}-\mathbf {R} _{n'})h_{l}^{+}(\kappa |\mathbf {R} _{n}-\mathbf {R} _{n'}|)\right](1-\delta _{nn'})\\C_{LL'L_{1}}&=\int Y_{L}(\mathbf {q} )Y_{L'}(\mathbf {q} )Y_{L_{1}}(\mathbf {q} )\,d\Omega _{q}\end{aligned}}}$

${\displaystyle T_{nn'}^{LL'}(\kappa )=\{[\tau ^{-1}(\kappa )-B(\kappa )]^{-1}\}_{nn'}^{LL'}}$

${\displaystyle {\begin{aligned}D(E)-D_{0}(E)&={\frac {2}{N\pi }}\mathrm {Im\,Tr} {\frac {d}{dE}}\ln[T(\kappa )]\\&=-{\frac {2}{N\pi }}\mathrm {Im\,Tr} {\frac {d}{dE}}\ln[\tau ^{-1}-B(\kappa )]\\&=-{\frac {2}{N\pi }}\mathrm {Im\,Tr} \left[T(\kappa )\left({\frac {d\tau ^{-1}}{dE}}-{\frac {dB(\kappa )}{dE}}\right)\right]\\&=-{\frac {2}{N\pi }}\mathrm {Im} \sum _{n,L}\sum _{n_{1},L_{1}}T_{nn_{1}}^{LL_{1}}\left[\delta _{{n_{1}}n}^{{L_{1}}L}{\frac {d(\tau _{n}^{l}(\kappa ))^{-1}}{dE}}-{\frac {d}{dE}}B_{{n_{1}}n}^{{L_{1}}L}(\kappa )\right]\end{aligned}}}$

${\displaystyle D(E)=-{\frac {1}{\pi }}\mathrm {Im} \,G(E)=-{\frac {1}{\pi }}\mathrm {Im} \,\{{\tilde {G}}(E)+{\tilde {G}}(E)T(E){\tilde {G}}(E)\}}$

${\displaystyle {\begin{aligned}D(E)=-{\frac {1}{\pi }}\mathrm {Im} \,G(E)&=-{1 \over {\pi }}\mathrm {Im} \,\{G_{0}(E)+G_{0}(E)\,T(E)\,G_{0}(E)\}\\&=D_{0}(E)-{\frac {1}{\pi }}\mathrm {Im} \,G_{0}(E)\,T(E)\,G_{0}(E)\end{aligned}}}$