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 ${G}₀⁺(z{-}{z}^{{'}}, k̄n{ω})={-}ia{Σ}{j}^{}{{e}^{i{k}ⱼ}(z{-}{z}^{{'}})}{v({k}ⱼk̄n)} sgn (z{-}{z}^{{'}})+{G}BC$, where $k̄=({k}ₓ,{k}y)$, $n$ 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 ${ε}({k}zk̄n)$ is equal to the parameter ${ω}$, and for which $v({k}ⱼk̄n)sgn(z{-}{z}^{{'}})>0$, if ${k}ⱼ$ is real, or $Im{k}ⱼsgn(z{-}{z}^{{'}})>0$, if ${k}ⱼ$ is complex. ${G}BC$ represents integrals around branch cuts, $a$ is the height of a unit cell, and $v({k}zk̄n){≡}{{∂}{ε}({k}zk̄n)}{{∂}{k}z}$. The above expression can be regarded as a generalization of the usual one-dimensional Green's function of quantum mechanics. ${G}₀⁺({ω})$ diverges whenever ${ω}$ is such that some $v({k}ⱼk̄n)$ goes to zero, and as a result the generalized phase shift ${η}({ω}k̄)$ has discontinuities of ${-}{{π}}{2}$ at these values of ${ω}$. 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 ${{↔}}{M}=0$, where ${{↔}}{M}$ 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 ${ω}=k̄=0$, however, because of a restriction on the force constants. The significance of ${η}({ω}k̄)$ is that the surface or interface density of states ${Δ}{ρ}({ω}k̄)$ is given by ${{π}}^{{-}1}{{∂}{η}({ω}k̄)}{{∂}{ω}}$. Each discontinuity of ${-}{{π}}{2}$ in ${η}({ω}k̄)$ at an extremum ${{ω}}₀$ thus produces a contribution ${-}{{δ}({ω}{-}{{ω}}₀)}{2}$ in ${Δ}{ρ}({ω}k̄)$.