Green functions complex energy

Fig. 2. Contour for integration in the complex energy plane Z. As any retarded Green s function is analytic in the upper half-plane Z, the analytic continuation GR(E) — Gr(Z), ReZ = E, and ImZ > 0 is well defined. The semi-circle C and the real line (HR> Hl) are used to calculate p = f dE G<(E)/27 i if fx — ir = eV > 0.

This Green function is analytic in the complex energy plane except for the bound-state poles at En, with branch points at = 1 and cuts along the real axis for E > 1. Bound states occur only at energies E > 0. The firee-photon propagator appears as a time-ordered product of firee-photon field operators (in Feynman gauge)... 

The complex energy, 2 of Eq. (5) is normally understood in fhe context of resonance scattering theory as the complex pole in the Breit-Wigner amplitude, or in the S-matrix, or in the optical potential of Feshbach s fheory," or in the Green s function, e.g.. Refs. [2,6-8]. 

The Green s function (4.60) has, for real v, a branch cut along the real E-axis, thus exhibiting a completely continuous spectrum. The spectral density function can be determined as discussed before. The complex energy variable is written d E = t + ip and the real quantity... 

Let z be a complex variable in the energy plane. Then the Green s functions corresponding to HB0 and Hel are, respectively,... 

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