Dyson integral equation

The detailed diagrammatic analysis results in the following Dyson integral equation for the mixed Keldysh function on the Keldysh contour [99] ... 

The lesser (retarded, advanced) Green function matrix of a nonequilibrium molecule G<(ll A> = GAjft A> can be found from the Dyson-Keldysh equations in the integral form... 

If we could obtain G( ), or at least a better approximation to it than Go( ), then we would be able to improve upon Koopmans theorem IP s and EA s while retaining the one-particle picture associated with HF theory. At first glance it does not appear possible to construct an exact one-particle theory since the many-electron Hamiltonian contains two-particle interactions. F. Dyson surmounted this apparent difficulty by introducing an effective potential which was energy dependent, called the self-energy. Moreover, he showed that the exact G( ) obeys the integral equation (now called the Dyson equation) ... 

Equation (2.44) is a complicated integral equation connecting the KS Green s function, the xc-component of the full self-energy, the xc-potential and the full Green s function. Does this relation have an3dhing to do with the 0PM equation (2.27) The first step towards an answer to this question is provided by a repeated use of the Dyson equation (2.43). After insertion of (2.43) the... 

Equation (8) is a type of Dyson equation known as the Chandler-Andersen (oh RISM — reference interaction site method") equation (Chandler, 1982 Chandler and Andersen, 1972). It is a generalization of the Ornstein-Zernike integral equation for simple atomic fluids (Hansen and McDonald, 1986). It is an integral equation which relates the unknown h ( r-r ) to the equally unknown c ( r-r ). Indeed, Eq. (8) is essentially a definition of c yir). Another relationship is required to close the equation, and to construct a closure a field theoretic perspective can be suggestive. We turn to such a perspective now. 

The usual procedure to evaluate the Green s function is by use of a perturbation expansion. Dyson showed that the exact G(IP) obeys the following integral equation (now called the Dyson equation) ... 

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 ... 

A simple repetition of the iteration procedure (2.20)-(2.22) results in divergence of higher order solutions. However, a perturbation theory series may be summed up so that all unbound diagrams are taken into account, just as is usually done for derivation of the Dyson equation [120]. As a result P satisfies the integral-differential equation... 

In the same way, we obtain the integral Dyson equation for the QD Keldysh functions (52) ... 

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