Vol. 2, No. 3, 2020

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Mean-field model for the junction of two quasi-1-dimensional quantum Coulomb systems

Ling-Ling Cao

Vol. 2 (2020), No. 3, 533–580
Abstract

Junctions appear naturally when one studies surface states or transport properties of quasi-1-dimensional materials such as carbon nanotubes, polymers and quantum wires. These materials can be seen as 1-dimensional systems embedded in the 3-dimensional space. We first establish a mean-field description of reduced Hartree–Fock-type for a 1-dimensional periodic system in the 3-dimensional space (a quasi-1-dimensional system), the unit cell of which is unbounded. With mild summability condition, we next show that a quasi-1-dimensional quantum system in its ground state can be described by a mean-field Hamiltonian. We also prove that the Fermi level of this system is always negative. A junction system is described by two different infinitely extended quasi-1-dimensional systems occupying separate half-spaces in three dimensions, where coulombic electron-electron interactions are taken into account and without any assumption on the commensurability of the periods. We prove the existence of the ground state for a junction system, the ground state is a spectral projector of a mean-field Hamiltonian, and the ground state density is unique.

Keywords
mean field, junction, quasi-1-dimensional, quantum Coulomb system
Mathematical Subject Classification 2010
Primary: 49S05, 58E99
Milestones
Received: 24 April 2019
Revised: 26 March 2020
Accepted: 10 May 2020
Published: 17 November 2020
Authors
Ling-Ling Cao
Université Paris-Est Marne-la-Vallée, CERMICS (ENPC)
Marne-la-Vallée
France