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Développement de modèles macroscopiques pour des systèmes quantiques non linéaires hors équilibre

Abstract : This document proposes a mathematical framework to analyze far-from-equilibrium electronic transport in mesoscopic devices, like heterostructures or superlattices. This leads to study the asymptotics of nonlinear-1D-Schrödinger-Poisson systems. The potential is made with quantum wells in a semiclassical island with cliffs. The main result for the nonlinear theory is the existence of asymptotical steady states. Moreover we show that they lie in a finite-dimensional subspace of continuous functions. Besides, one focuses on the understanding of spectral properties of the linear Schrödinger Operator for this semilinear problem. In this framework we lead the analysis over the continuous spectrum. The quantum wells generate quantum resonances. First we give results about functions of the Hamiltonian. Next, we focus on the more delicate case of functions of the asymptotical momentum. Results are given when the distribution of the wells over the island ensures good treatement of the resonances : for instance, whenever the wells are gathered or confined far from the boundary of the island. This establishes classical-like solutions, and we finish our analysis by showing the possible existence of quantum-like solutions.
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Contributor : Mamodyasine Patel <>
Submitted on : Thursday, February 3, 2005 - 2:05:45 PM
Last modification on : Thursday, January 7, 2021 - 4:12:35 PM
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  • HAL Id : tel-00008345, version 1


Mamodyasine Patel. Développement de modèles macroscopiques pour des systèmes quantiques non linéaires hors équilibre. Mathématiques [math]. Université Rennes 1, 2005. Français. ⟨tel-00008345⟩



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