Solution to the many-body problem in one point

In this work we determine the one-body Green s function as solution of a set of functional integro-differential equations, which relate the one-particle Green s function to its functional derivative with respect to an external potential. In the same spirit as Lani et al (2012 New J. Phys. 14 [http://dx.doi.org/10.1088/1367-2630/14/1/013056] 013056 ), we do this in a one-point model, where the equations become ordinary differential equations (DEs) and, hence, solvable with standard techniques. This allows us to analyze several aspects of these DEs as well as of standard methods for determining the one-body Green s function that are important for real systems. In particular: (i) we present a strategy to determine the physical solution among the many mathematical solutions; (ii) we assess the accuracy of an approximate DE related to the $\#$$\#$IMG$\#$$\#$ [http://ej.iop.org/images/1367-2630/16/11/113025/njp503025ieqn1.gif] $GW$ +cumulant method by comparing it to the exact physical solution and to standard approximations such as $\#$$\#$IMG$\#$$\#$ [http://ej.iop.org/images/1367-2630/16/11/113025/njp503025ieqn2.gif] $GW$ ; (iii) we show that the solution of the approximate DE can be improved by combining it with a screened interaction in the random-phase approximation. (iv) We demonstrate that by iterating the $\#$$\#$IMG$\#$$\#$ [http://ej.iop.org/images/1367-2630/16/11/113025/njp503025ieqn3.gif] $GW$ Dyson equation one does not always converge to a $\#$$\#$IMG$\#$$\#$ [http://ej.iop.org/images/1367-2630/16/11/113025/njp503025ieqn4.gif] $GW$ solution and we discuss which iterative scheme is the most suitable to avoid such errors.