Green's function and generalized phase shift for surface and interface problems

For electrons, phonons, etc., and regardless of symmetry, the Green's function in any mixed Wannier-Bloch representation is $G_0^{+}(z-z^{\prime }, \overline{\mathrm{k}}n\omega )=-ia\Sigma j^{}\frac{e^{i{k}_j}(z-z^{\prime })}{v\left(k_j\overline{\mathrm{k}}n\right)} \mathrm{sgn} (z-z^{\prime })+G_{\mathrm{BC}}$, where $\overline{\mathrm{k}}=(k_x,k_y)$, $n$ is the branch index, and the values of $z$ correspond to lattice points. The $k_j$ are those values of $k_z$ for which the eigenvalue $\epsilon \left(k_z\overline{\mathrm{k}}n\right)$ is equal to the parameter $\omega$, and for which $v\left(k_j\overline{\mathrm{k}}n\right)\mathrm{sgn}(z-z^{\prime })>\mathrm{}0\mathrm{}$, if $k_j$ is real, or $\mathrm{Im}{k}_j\mathrm{sgn}(z-z^{\prime })>\mathrm{}0\mathrm{}$, if $k_j$ is complex. $G_{\mathrm{BC}}$ represents integrals around branch cuts, $a$ is the height of a unit cell, and $v\left(k_z\overline{\mathrm{k}}n\right)\equiv \frac{\partial \epsilon \left(k_z\overline{\mathrm{k}}n\right)}{\partial k_z}$. The above expression can be regarded as a generalization of the usual one-dimensional Green's function of quantum mechanics. $G_0^{+}\left(\omega \right)$ diverges whenever $\omega$ is such that some $v\left(k_j\overline{\mathrm{k}}n\right)$ goes to zero, and as a result the generalized phase shift $\eta \left(\omega \overline{\mathrm{k}}\right)$ has discontinuities of $-\frac{\pi }{2}$ at these values of $\omega$. These discontinuities are present regardless of the strength of $V$, the perturbation associated with creating a pair of surfaces or interfaces. There is an exception: If det $\overleftrightarrow{M}=0\mathrm{}$, where $\overleftrightarrow{M}$ is a matrix defined in terms of the matrix elements of $V$, then the discontinuity is eliminated. This condition is analogous to that for a "zero-energy resonance" in $s$-wave potential scattering, and it will ordinarily occur only at particular transitional strengths of $V$. The condition is always satisfied for acoustic phonons at $\omega =\overline{\mathrm{k}}=0\mathrm{}$, however, because of a restriction on the force constants. The significance of $\eta \left(\omega \overline{\mathrm{k}}\right)$ is that the surface or interface density of states $\Delta \rho \left(\omega \overline{\mathrm{k}}\right)$ is given by ${\pi }^{-1\mathrm{}}\frac{\partial \eta \left(\omega \overline{\mathrm{k}}\right)}{\partial \omega }$. Each discontinuity of $-\frac{\pi }{2}$ in $\eta \left(\omega \overline{\mathrm{k}}\right)$ at an extremum ${\omega }_0$ thus produces a contribution $-\frac{\delta (\omega -{\omega }_0)}{2}$ in $\Delta \rho \left(\omega \overline{\mathrm{k}}\right)$.