Dyson’s equation

To construct the Greenian for the metal-support system described above, we start from the infinite Greenians for the metal and the semiconductor. Dyson s equation is used to obtain the Greenians for the finite or semi-infinite components, and again to glue the two components together to produce the Greenian for the composite system. 

Balabin lA, Onuchic JN. Connection between simple models and quantum chemical models for electron-transfer tunneling matrix element calculations A Dyson s equations-based approach. J. Phys. Chem. 1996 100 11573-11580. 

Caianiello, E.R. (1953). On the Quantum Field Theory. I. Explicit Solution of Dyson s Equation in Electrodynamics Without Use of Feynman Graphs. Nuovo Cimento, 10,1634-1652. 

One way to obtain the single-particle Green s function is via solution of what is called Dyson s equation [21, 48, 49],... 

The single-particle Green s function is related to the irreducible self energy by Dyson s equation (38) and to the two-particle Green s function by the equation of motion (37). It can also be related to the xc potential of DFT by the Sham-Schliiter equation [50]... 

All three single-particle equations, Hartree, Hartree-Fock and Kohn-Sham can also be interpreted as approximations to Dyson s equation (38), which can be rewritten as [48]... 

The connection between these quantities is provided by Dyson s equations (see e.g.[125]),... 

Due to this form invariance of the Lagrangian under the renormalization prescription the stracture of the theory, which e.g. expresses itself in Dyson s equations and Ward-Takahashi identities, remains completely unchanged. The renormalized Greens functions, i.e. the vacuum expectation values of f and A, are thus simply obtained from... 

This is useful if we know the eigenfunctions of //o> but wish to calculate the solutions for H Dyson s equations are ... 

Many of the algebraic properties of the single-particle Green s function, in particular Dyson s equation, are transferable to two-particle propagators if they are constructed starting from an orthonormal set of primary states. In this paper we will construct propagators Q(u)) with the help of the extended... 

This equation is the main result of the present considerations. In order to define the two-particle self energy (w) and for establishing the connection to the familiar form of Dyson s equation we adopt a perturbation theoretical view where a convenient single-particle description (e. g. the Hartree-Fock approximation) defines the zeroth order. We will see later that the coupling blocks and vanish in a single-particle approximation. Consequently the extended Green s function is the proper resolvent of the zeroth order primary block which can be understood as an operator in the physical two-particle space ... 

The dynamic part DJ E), which can be found in the literature [8], possesses poles related to resonances of two-particle-hole and higher excitations and a branch cut above the ionisation threshold. It has an analytic structure similar to the dynamic self energy (42) from the formal point of view. This is not surprising since the projection method for deriving the Feshbach Hamiltonian is formally similar to the derivation of Dyson s equation of Sec. Ill or in Refs. [26,9]. There, a partitioning was performed yielding a projection to the primary space spanned by the orthonormal T-states while Feshbach, on the other hand, projected onto the space spanned by the non-orthonormal states aj 5 ). 

Together with eq. (A7) this defines the dispersion of collective quadrupolar excitations that is given by the zeros of the denominator in eq. (A7). The renormalized phonon propagator is now obtained by Dyson s equation (fig. 59a)... 

Here, Q t, r) denotes the Green s function for the reduced Liouvillian C t) = Cs + Csf(t) in the presence of external driving field. Denote also Qs t, r) = -%Cs t-T) qj. field-free Green s function. Using Dyson s equation,... 

Dyson s equation defines the Green matrix for the system with a perturbation in the form... 

Note that G/ r refers to the full one-particle Green s function. Eq. (2.10) is an integral equation, but if the kernel K(iA> iAL-Uk) depends only on CJ> the Bethe-Salpeter equation would reduce to a Dyson s equation ... 

To perform these calculations one needs the self-energy structure of the one-particle Green s function. The general structure of the one-particle Green s function presented in Eg. (2.23) is given by the Dyson s equation... 

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