Green's functions for surface physics

The following theorem is proved for a partial differential eigenvalue equation in a periodic system: $\ensuremath{-}2\ensuremath{\pi}i\ensuremath{\int}{\mathrm{sj}}^{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}}_{0}[{\overline{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n), \ensuremath{\psi}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}}{n}^{\ensuremath{'}})]\ifmmode\cdot\else\textperiodcentered\fi{}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}={v}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n){\ensuremath{\delta}}_{\mathrm{kjk}{j}^{\ensuremath{'}}}{\ensuremath{\delta}}_{n{n}^{\ensuremath{'}}}$, if $\ensuremath{\epsilon}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{\ensuremath{'}}{n}^{\ensuremath{'}})=\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)$ and ${{k}^{\ensuremath{'}}}_{i}={k}_{i}$ for $i\ensuremath{\ne}j$. Here $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)$ is an eigenvalue, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)$ is an eigenfunction, and $\overline{\ensuremath{\psi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)$ is a solution to the adjoint eigenvalue problem satisfying $\ensuremath{\int}{\mathrm{cell}}^{}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}{\overline{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}{n}^{\ensuremath{'}})={\ensuremath{\delta}}_{n{n}^{\ensuremath{'}}}$. Also, ${v}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)\ensuremath{\equiv}\frac{\ensuremath{\partial}\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)}{\ensuremath{\partial}{k}_{j}}$, ${s}_{j}$ is a cross section of the unit cell, and ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}}_{0}$ is the bilinear concomitant. The above theorem is used to evaluate the bulk Green's function in closed form: ${G}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}};\overline{k}\ensuremath{\epsilon})=\ensuremath{-}2\ensuremath{\pi}i{\ensuremath{\Sigma}{\mathrm{nl}}^{}[\frac{\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};\overline{k}{k}_{\mathrm{nl}}^{+}n){\overline{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}};\overline{k}{k}_{\mathrm{nl}}^{+}n)}{ {v}_{3}(\overline{k}{k}_{\mathrm{nl}}^{+}n)}]\ensuremath{\theta}({x}_{3}\ensuremath{-}{{x}_{3}}^{\ensuremath{'}})\ensuremath{-}\ensuremath{\Sigma}{\mathrm{nl}}^{}[\frac{\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};\overline{k}{k}_{\mathrm{nl}}^{\ensuremath{-}}n){\overline{\ensuremath{\psi}}}^{\ifmmode\dagger\else\textdagger\fi{}} ({\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}};\overline{k}{k}_{\mathrm{nl}}^{\ensuremath{-}}n)}{{v}_{3}(\overline{k}{k}_{\mathrm{nl}}^{\ensuremath{-}}n)}] \ensuremath{\theta}({{x}_{3}}^{\ensuremath{'}}\ensuremath{-}{x}_{3})}$, where $\overline{k}\ensuremath{\equiv}({k}_{1},{k}_{2})$. The ${k}_{\mathrm{nl}}$ are those values of ${k}_{3}$ for which $\ensuremath{\epsilon}(\overline{k}{k}_{\mathrm{nl}}n)=\ensuremath{\epsilon}$, partitioned into the two sets ${k}_{\mathrm{nl}}^{+}$ and ${k}_{\mathrm{nl}}^{\ensuremath{-}}$ according to the boundary conditions on ${G}_{0}$. A Green's function $G(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}};\overline{k}\ensuremath{\epsilon})$ in the presence of an interface is given by the above expression if $\ensuremath{\psi}$ is replaced by $\ensuremath{\Psi}$, an eigenfunction that grows out of $\ensuremath{\psi}$ as the interface is approached. This expression also gives the exact many-body Green's function ${\mathcal{G}}_{0}$ (or $\mathcal{G}$) if $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)$ and $\ensuremath{\psi}$ (or $\ensuremath{\Psi}$) are interpreted as solutions to an eigenvalue problem involving the self-energy. Finally, the expression holds for nondifferential equations---e.g., the matrix eigenvalue equation for phonons or electrons in a localized representation; in this case, the derivation is based on the analytic properties of $\ensuremath{\epsilon}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)$ and $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}n)$ at complex $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$.