GREEN-FUNCTIONS FOR SURFACE PHYSICS

The following theorem is proved for a partial differential eigenvalue equation in a periodic system: -2 isjP0[»(kn), (kn)]dS=vj(kn)kjkj nn, if (k n)=(kn) and ki=ki for i j. Here (kn) is an eigenvalue, (k n) is an eigenfunction, and »(kn) is a solution to the adjoint eigenvalue problem satisfying celldx»(x;kn)(x;kn)=nn. Also, vj(kn)(kn)kj, sj is a cross section of the unit cell, and P0 is the bilinear concomitant. The above theorem is used to evaluate the bulk Green's function in closed form: G0(xx;k»)=-2i{nl[(x;k»knl+n) »(x;k»knl+n) v3(k»knl+n)](x3-x3)-nl[(x;k»knl-n)»(x;k»knl-n) v3(k»knl-n)] (x3-x3)}, where k»(k1,k2). The knl are those values of k3 for which (k»knln)=, partitioned into the two sets knl+ and knl- according to the boundary conditions on G0. A Green's function G(xx;k») in the presence of an interface is given by the above expression if is replaced by , an eigenfunction that grows out of as the interface is approached. This expression also gives the exact many-body Green's function G0 (or G) if (kn) and (or) 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 (kn) and (kn) at complex k. © 1979 The American Physical Society.