Green's functions for surface physics

The following theorem is proved for a partial differential eigenvalue equation in a periodic system: $-2\mathrm{}\pi i\int {\mathrm{sj}}^{}{\overrightarrow{\mathrm{P}}}_0\left[{\overline{\psi }}^{†}\right(\overrightarrow{\mathrm{k}}n), \psi ({\overrightarrow{\mathrm{k}}}^{\prime }{n}^{\prime }\left)\right]·d\overrightarrow{\mathrm{S}}=v_j\left(\overrightarrow{\mathrm{k}}n\right){\delta }_{\mathrm{kjk}{j}^{\prime }}{\delta }_{n{n}^{\prime }}$, if $\epsilon \left({\overrightarrow{\mathrm{k}}}^{\prime }{n}^{\prime }\right)=\epsilon \left(\overrightarrow{\mathrm{k}}n\right)$ and ${k^{\prime }}_i=k_i$ for $i\ne j$. Here $\epsilon \left(\overrightarrow{\mathrm{k}}n\right)$ is an eigenvalue, $\psi \left(\overrightarrow{\mathrm{k}}n\right)$ is an eigenfunction, and $\overline{\psi }\left(\overrightarrow{\mathrm{k}}n\right)$ is a solution to the adjoint eigenvalue problem satisfying $\int {\mathrm{cell}}^{}d\overrightarrow{\mathrm{x}}{\overline{\psi }}^{†}(\overrightarrow{\mathrm{x}};\overrightarrow{\mathrm{k}}n)·\psi (\overrightarrow{\mathrm{x}};\overrightarrow{\mathrm{k}}{n}^{\prime })={\delta }_{n{n}^{\prime }}$. Also, $v_j\left(\overrightarrow{\mathrm{k}}n\right)\equiv \frac{\partial \epsilon \left(\overrightarrow{\mathrm{k}}n\right)}{\partial k_j}$, $s_j$ is a cross section of the unit cell, and ${\overrightarrow{\mathrm{P}}}_0$ is the bilinear concomitant. The above theorem is used to evaluate the bulk Green's function in closed form: $G_0(\overrightarrow{\mathrm{x}}{\overrightarrow{\mathrm{x}}}^{\prime };\overline{k}\epsilon )=-2\mathrm{}\pi i\{\Sigma {\mathrm{nl}}^{}\left[\frac{\psi (\overrightarrow{\mathrm{x}};\overline{k}{k}_{\mathrm{nl}}^{+}n){\overline{\psi }}^{†}({\overrightarrow{\mathrm{x}}}^{\prime };\overline{k}{k}_{\mathrm{nl}}^{+}n)}{ v_3\left(\overline{k}{k}_{\mathrm{nl}}^{+}n\right)}\right]\theta (x_3-{x_3}^{\prime })-\Sigma {\mathrm{nl}}^{}\left[\frac{\psi (\overrightarrow{\mathrm{x}};\overline{k}{k}_{\mathrm{nl}}^{-}n){\overline{\psi }}^{†} ({\overrightarrow{\mathrm{x}}}^{\prime };\overline{k}{k}_{\mathrm{nl}}^{-}n)}{v_3\left(\overline{k}{k}_{\mathrm{nl}}^{-}n\right)}\right] \theta ({x_3}^{\prime }-x_3)\}$, where $\overline{k}\equiv (k_1,k_2)$. The $k_{\mathrm{nl}}$ are those values of $k_3$ for which $\epsilon \left(\overline{k}{k}_{\mathrm{nl}}n\right)=\epsilon$, partitioned into the two sets $k_{\mathrm{nl}}^{+}$ and $k_{\mathrm{nl}}^{-}$ according to the boundary conditions on $G_0$. A Green's function $G(\overrightarrow{\mathrm{x}}{\overrightarrow{\mathrm{x}}}^{\prime };\overline{k}\epsilon )$ in the presence of an interface is given by the above expression if $\psi$ is replaced by $\Psi$, an eigenfunction that grows out of $\psi$ as the interface is approached. This expression also gives the exact many-body Green's function ${\mathcal{G}}_0$ (or $\mathcal{G}$) if $\epsilon \left(\overrightarrow{\mathrm{k}}n\right)$ and $\psi$ (or $\Psi$) 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 $\epsilon \left(\overrightarrow{\mathrm{k}}n\right)$ and $\psi \left(\overrightarrow{\mathrm{k}}n\right)$ at complex $\overrightarrow{\mathrm{k}}$.