ABSTRACT
G(z) = (zI −H)−1 , z = + iδ , G (z∗) = G (z)† , (4.1) where I is the unity operator. Any representation of such a resolvent is called a Green’s functions, e.g., also the following configuration space representation of G(z),
< r |G(z)| r0 > = G(r, r0; z) . (4.2) The so-called side-limits of G(z) are then defined by
G(z) = ⎧ ⎨ ⎩ G+() ; δ > 0
G−() ; δ < 0 , (4.3)
G+() = G−()† , (4.4) and therefore lead to the property,
ImG+() = 1 2i ¡G+()− G−()¢ , (4.5)
in
or — assuming for matters of simplicity only a discrete eigenvalue spectrum {k} of H,
ImTrG±() = ∓π−1 X k
δ(− k) , (4.6)
n() = ∓ImTrG±() , (4.7) where Tr denotes the trace of an operator and n() is the density of states.
