Green's functions for surface physics

The following theorem is proved for a partial differential eigenvalue equation in a periodic system: ${-}2{π}i{∫}{sj}^{}{{{→}}{P}}₀[{{{ψ}}}^{{}}({{→}}{k}n), {ψ}({{{→}}{k}}^{{'}}{n}^{{'}})]·{}d{{→}}{S}={v}ⱼ({{→}}{k}n){{δ}}_{kjk{j}^{{'}}}{{δ}}_{n{n}^{{'}}}$, if ${ε}({{{→}}{k}}^{{'}}{n}^{{'}})={ε}({{→}}{k}n)$ and ${{k}^{{'}}}ᵢ={k}ᵢ$ for $i{≠}j$. Here ${ε}({{→}}{k}n)$ is an eigenvalue, ${ψ}({{→}}{k}n)$ is an eigenfunction, and ${{ψ}}({{→}}{k}n)$ is a solution to the adjoint eigenvalue problem satisfying ${∫}{cell}^{}d{{→}}{x}{{{ψ}}}^{{}}({{→}}{x};{{→}}{k}n)·{}{ψ}({{→}}{x};{{→}}{k}{n}^{{'}})={{δ}}_{n{n}^{{'}}}$. Also, ${v}ⱼ({{→}}{k}n){≡}{{∂}{ε}({{→}}{k}n)}{{∂}{k}ⱼ}$, ${s}ⱼ$ is a cross section of the unit cell, and ${{{→}}{P}}₀$ is the bilinear concomitant. The above theorem is used to evaluate the bulk Green's function in closed form: ${G}₀({{→}}{x}{{{→}}{x}}^{{'}};k̄{ε})={-}2{π}i{{Σ}{nl}^{}[{{ψ}({{→}}{x};k̄{k}ₙₗ⁺n){{{ψ}}}^{{}}({{{→}}{x}}^{{'}};k̄{k}ₙₗ⁺n)}{ {v}₃(k̄{k}ₙₗ⁺n)}]{θ}({x}₃{-}{{x}₃}^{{'}}){-}{Σ}{nl}^{}[{{ψ}({{→}}{x};k̄{k}ₙₗ^{{-}}n){{{ψ}}}^{{}} ({{{→}}{x}}^{{'}};k̄{k}ₙₗ^{{-}}n)}{{v}₃(k̄{k}ₙₗ^{{-}}n)}] {θ}({{x}₃}^{{'}}{-}{x}₃)}$, where $k̄{≡}({k}₁,{k}₂)$. The ${k}ₙₗ$ are those values of ${k}₃$ for which ${ε}(k̄{k}ₙₗn)={ε}$, partitioned into the two sets ${k}ₙₗ⁺$ and ${k}ₙₗ^{{-}}$ according to the boundary conditions on ${G}₀$. A Green's function $G({{→}}{x}{{{→}}{x}}^{{'}};k̄{ε})$ in the presence of an interface is given by the above expression if ${ψ}$ is replaced by ${Ψ}$, an eigenfunction that grows out of ${ψ}$ as the interface is approached. This expression also gives the exact many-body Green's function ${G}₀$ (or $G$) if ${ε}({{→}}{k}n)$ and ${ψ}$ (or ${Ψ}$) 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 ε(→kn) and ψ(→kn) at complex →k.