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Junctions appear naturally when one studies surface states or transportproperties of quasi one dimensional materials such as carbon nanotubes,polymers and quantum wires. These materials can be seen as 1D systems embeddedin the 3D space. In this article, we first establish a mean--field descriptionof reduced Hartree--Fock type for a 1D periodic system in the 3D space (a quasi1D system), the unit cell of which is unbounded. With mild summabilitycondition, we next show that a quasi 1D system in its ground state can bedescribed by a mean--field Hamiltonian. We also prove that the Fermi level ofthis system is always negative. A junction system is described by two differentinfinitely extended quasi 1D systems occupying separately half spaces in 3D,where Coulombic electron-electron interactions are taken into account andwithout any assumption on the commensurability of the periods. We prove theexistence of the ground state for a junction system, the ground state is aspectral projector of a mean--field Hamiltonian, and the ground state densityis unique.

Mean--field stability for the junction of quasi 1D systems with Coulomb interactions