Green's functions for surface physics
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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)](x3x3)nl[(x;k»knln)»(x;k»knln) v3(k»knln)] (x3x3)}, 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 manybody Green's function G0 (or G) if (kn) and (or) are interpreted as solutions to an eigenvalue problem involving the selfenergy. 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.
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http://dx.doi.org/10.1103/PhysRevB.20.1454
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