A Dyson Equation for Non-Equilibrium Green's Functions in the Partition-Free Setting

We consider a small interacting sample coupled to several non-interacting leads. Initially, the system is at thermal equilibrium. At some instant t 0 the system is set into the so called partition-free transport scenario by turning on a bias on the leads. Using the theory of Volterra operators we rigorously formulate a Dyson equation for the retarded Green's function and we establish a closed formula for the associated proper interaction self-energy.