Ebert - Psi-k

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Ludwig Maximilians- Universität München Spectral representation of G(⃗r, ⃗r ′ , E) ψ i (⃗r, E i ) complete and orthogonal set of eigenfuctions (E i − H) ψ i (⃗r, E i ) = 0 (E − H) G(⃗r, ⃗r ′ , E) = δ(⃗r − ⃗r ′ ) G(⃗r, ⃗r ′ , E) for fixed ⃗r ′ and E can be expanded into a series of ψ i (⃗r, E i ) (Mercer’s theorem) ∑ G + (⃗r, ⃗r ′ , E) = lim ɛ→0 retarded Green’s function i ψ i (⃗r)ψ × i (⃗r ′ ) E − E i + iɛ poles of G + (⃗r, ⃗r ′ , E) indicate eigenvalues lim ɛ→0 avoids divergencies NB: × is in general equivalent to † KKR07 – p.8/100
Ludwig Maximilians- Universität München Densities and expectation values The Dirac identity lim ɛ→0 I 1 x + iɛ = −π δ(x) leads to − 1 π IG+ (⃗r, ⃗r ′ , E) = ∑ i ψ i (⃗r)ψ × i (⃗r ′ )δ(E − E i ) IG + (⃗r, ⃗r ′ , E) can be interpreted as density matrix Density of states n(E) = − 1 π I ∫ V d 3 r G + (⃗r, ⃗r, E) Particle density n(⃗r) = − 1 π I ∫ EFdE G + (⃗r, ⃗r, E) Expectation value 〈A〉 = − 1 π I ∫ EFdE ∫ V d 3 r A G + (⃗r, ⃗r, E) KKR07 – p.9/100
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