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Quantum hypernetted chain approximation for one dimensional fermionic systems

In this comprehensible article we develop, following Fantoni and Rosati formalism, a hypernetted chain approximation for one dimensional systems of fermions. Our scheme differs from previous treatments in the form that the whole set of diagrams is grouped: we do it in terms of non-nodal, non-composite and elementary graphs. This choice makes the deduction of equations more transparent. Equations for the pair distribution functions of one component systems as well as binary mixtures are obtained. We apply they to experimentally realizable quasi-one dimensional systems, the so called quantum wires which we model, within Sommerfeld-Pauli spirit, as a 1D electron gas or as an electron-hole mixture. In order to use our one-dimensional equations we consider pair potentials that depend on the wires width.