Asymptotic quantum transport models of a two-dimensional electron gas are presented. The starting point is a singular perturbation of the threedimensional Schrödinger-Poisson system. The small parameter ε is the scaled width of the electron gas and appears as the lengthscale on which a one dimensional confining potential varies. The rigorous ε→ 0 limit is performed by projecting the three dimen...

In a previous work [N. Ben Abdallah, C. Negulescu, A one-dimensional transport model with small coherence lengths, Transp. Theory and Stat. Phys. 31, no. 4-6 (2002), 559-578] a one dimensional transport model accounting for both, quantum phenomena and smallness of coherence lengths, has been analyzed. In the limit of infinitely small coherence lengths, this model leads to the classical Vlasov e...

We consider the semiclassical limit for the nonlinear Schrödinger equation. We introduce a phase/amplitude representation given by a system similar to the hydrodynamical formulation, whose novelty consists of including some asymptotically vanishing viscosity. We prove that the system is always locally well-posed in a class of Sobolev spaces, and globally well-posed for a fixed positive Planck c...

Parasupersymmetry of the one-dimensional time-dependent Schrödinger equation is established. It is intimately connected with a chain of the time-dependent Darboux transformations. As an example a parasupersymmetric model of nonrelativistic free particle with threefold degenerate discrete spectrum of an integral of motion is constructed. 1. Supersymmetric quantum mechanics originally introduced ...