where where Σ( z ) is the self-energy, defined by Note that, by introducing the density of states ρ ( E ) = ( ∂E/∂k ) − 1 and letting V ( E ) = ρ ( E ) | v ( E ) | 2 ( V ( E )=0 for E > E$_{2}$ and E < E$_{1}$ ), the self-energy can be written in the equivalent form P denotes the principal value, and E is real-valued. Therefore, Σ( z ) has a branch cut in the interval ( E$_{1}$, E$_{2}$ ). For two assigned functions | χ 〉 = χ$_{a}$ | a 〉 + ∫ dkχ ( k ) | k 〉 and | ϕ 〉 = ϕ$_{a}$ | a 〉 + ∫ dkϕ ( k ) | k 〉 of the Hilbert space, the complex function G $_{χ,ϕ}$( z ) = 〈 χ | G ( z ) ϕ 〉 can be readily calculated as Δ( E ) = P ∫ E 2 E 1 dE V ( E ) E − E , (13) Σ( z ) = ∫ dk | v ( k ) | 2 z − E ( k ) . (10) Σ( z ) = ∫ E 2 E 1 dE V ( E ) z − E . (11) Σ( z = E ± i 0 $^{+}$) = Δ( E ) ∓ iπV ( E ) (12) Note also that Σ( z ) is not defined on the segment ( E$_{1}$, E$_{2}$ ) of the real axis, and one has Substitution of Eqs.(6-9) into Eq.(14) finally yields where we have set G $_{χ,ϕ}$( z ) = χ ∗ $_{a}$ϕ$_{a}$ G $_{a,a}$( z ) + χ ∗ a ∫ dkϕ ( k ) G $_{a,k}$( z ) + ϕ$_{a}$ ∫ dkχ $^{∗}$( k ) G $_{k,a}$( z ) + ∫ dkdk $^{′}$χ$^{∗}$( k $^{′}$) ϕ ( k ) G k ′ $_{,k}$( z ) (14) G $_{χ,ϕ}$( z ) = G $_{a,a}$( z )Φ$_{1}$( z ) + Φ$_{2}$( z ) (15) Φ$_{1}$( z ) = χ ∗ $_{a}$ϕ$_{a}$ + χ ∗ a ∫ dE ρ ( E ) ϕ ( E ) v ( E ) z − E + ϕ$_{a}$ ∫ dE ρ ( E ) χ $^{∗}$( E ) v $^{∗}$( E ) z − E + + (∫ dE ρ ( E ) v ( E ) ϕ ( E ) z − E ) (∫ dE ρ ( E ) v $^{∗}$( E ) χ $^{∗}$( E ) z − E ) (16) Φ$_{2}$( z ) = ∫ dE ρ ( E ) χ $^{∗}$( E ) ϕ ( E ) z − E . (17) singularities. To this aim, let us notice that Φ$_{1}$( z ) and Φ$_{2}$( z ) are bounded functions of z and have a branch cut We are now ready to determine the singularities of the resolvent, i.e. the spectrum of H and possible spectral 3 G a,a ≡ 〈 a | G ( z ) a 〉 = 1 z − E$_{a}$ − Σ( z ) (6) G k,a ≡ 〈 k | G ( z ) a 〉 = ( z − E ( k ))( z − E$_{a}$ − Σ( z )) (8) G k,k ′ ≡ 〈 k | G ( z ) k $^{′}$〉 = v ( k $^{′}$) v $^{∗}$( k ) ( z − E ( k ))( z − E ( k $^{′}$))( z − E$_{a}$ − Σ( z )) + δ ( k − k $^{′}$) z − E ( k $^{′}$) (9) G a,k ≡ 〈 a | G ( z ) k 〉 = v ( k ) ( z − E ( k ))( z − E$_{a}$ − Σ( z )) (7) v $^{∗}$( k )