Poles of the Green’s functions

From (1.10), it follows that eigenvalues of the Hamiltonian superoperator correspond to poles of the Green s function, and therefore, to EADEs. Thus, we are faced with an eigenvalue problem [33,34],... 

The poles of the Green s function (6.68) determine the dispersion of the polariton in the region of the Fermi resonance. The energy of the polariton is determined from the condition A(E, k) = 0, which can be written in the form... 

It follows that the poles of the Green s function Goo z) determine the time dependence of an electron localized in the adsorbate orbital o We will compute Eq.(2.276c) in the limit of weak chemisorption fo(ty) and in the surface molecule limit... 

This means that the poles of the Green s function, i.e., the molecular orbital energies, are obtained as eigenvalues of a negative definite matrix with elements -Kr Srs + Srs)Ks. The molecular orbital coeflficients can straightforwardly be inferred from the residues at the poles of this spectral representation. 

Clearly only real values of E appear in this expression, but since the analytic continuation of the Green s function G+(E) into the lower half E-plane often has a pole in that region,... 

It follows from the expressions given above for the Green s functions G W(w), i = 1, 2,..., 6, that, when anharmonicity is characterized by the constants A and r, this leads to the appearance, along with poles of the type of eqn (6.2), of a new type of poles for the Green s function. These poles are determined by the equation... 

Fig. 12. Band structure for the PAM. The model parameters are the same as in fig. 11. However the features persist up to T/Tq 10. The solid line shows the real part of the Green s functions poles vs. e. the unrenotmalized band energy. The symbols ow the positions of the maxima in the f and d spectral functions. We characterize these peaks to be of f character whenever A (e, Bt)>Ad(ei, Ey) or d character...

For calculating bound-state energies and wave functions, A 0 and the filter becomes equivalent to the spectral density operator S Ei — H) [208]. In the case of resonances, the Green s function in Eq. (33) selects the contributions from those complex poles whose real parts lie near Ei. In what follows we consider only the filter defined in Eq. (33). 

First, let us consider field theory. If the Lagrangian theory defined in Chapter 11 is a genuine field theory, as we believe, the Green s function has a pole for k2 = - m2 (where m can be considered as the mass of the particle which the field describes) and also a cut for more negative values of k1. Thus, for large values of r, we must attribute a dominant role to this pole. At the pole, O(x) vanishes, and in the vicinity of this point we may set... 

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]. 

We have seen that in single-particle quantum mechanics the Green s function has poles at values of equal to the eigenvalues of the Hamiltonian. To generalize Green s function theory to many-particle systems, we first consider an independent particle description, such as the Hartree-Fock (HF),... 

The Green s function method described in the previous section is based on the observation that the time evolution of the initially prepared state s) is given by the Fourier transformation of the diagonal matrix element (s G s) ofthe Green s operator for the system G = (E—H + ie) and that the decay characteristic of this state is determined by the complex pole of this diagonal element ... 

The importance of the one-particle Green s function for the calculation of ionization and electron affinity spectra can already be appreciated from Eq. (1) regardless of sign, ionization energies and electron affinities relate to the poles of its first and second components, respectively. The associated residues correspond to... 

Clearly, the poles of GB0 and G determine the eigenvalues of HBO and Hel corresponding to bound states, while the branch cuts of Guo and G determine the eigenvalues corresponding to the continuous spectra of Hbo and Hel. It is convenient to relate GBO(z) to G(z). This relationship can be established by combining G(z) with a projection operator defined on the BO states. We proceed by defining three reduced Green s functions... 

As seen from Eq. (8), Im II(kw) is finite for k — 0, whereas cjk vanishes in this limit. Therefore the spin Green s function has a purely imaginary, diffusive pole near the T point, in compliance with the result of the hydrodynamic theory [12]. 

The resolvent in eq. (1.208) is called the one-electron Green s function and the notation for it reads G (z). The integration contour may be set in such a way that it encloses all the poles of the resolvent corresponding to the occupied MOs giving by this the required total projection operator. In the spin-orbital occupation number and the second quantization representations related to each other, one can write the operator projecting to the occupied (spin)-MO as an operator of the number of particles in it. Indeed, the expression... 

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