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Inflow boundary conditions for the time dependent one-dimensional Schrödinger equation



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S0764 4442 00 01759 6 FLA AID 1759 Vol 331 0 CRAcad 2000 06 15 Prn 27 11 2000 13 21 F PXMA1759 tex P 1 1 6 by Au p 1 C R Acad Sci Paris t 331 S rie I p 1 6 2000 Probl me math matiques de la m canique Mathematical Problems in Mechanics quations aux d riv es partielles Partial Differential Equations Inflow boundary conditions for the time dependent one dimensional Schr dinger equation Naoufel BEN ABDALLAH a Pierre DEGOND a Ir ne GAMBA b a Laboratoire de math matiques pour l indusrie et la physique UMR CNRS 5640 Universit Paul Sabatier 118 route de Narbonne 31062 Toulouse cedex 4 France E mail naoufel mip ups tlse fr degond mip ups tlse fr b Texas Institute for Computational and Applied Mathematics and Department of Mathematics University of Texas at Austin USA E mail gamba math utexas edu Re u le 3 juillet 2000 accept apr s r vision le 23 octobre 2000 Abstract A transient quantum model for one dimensional charge transport is derived and shown to lead in the semiclassical limit to the inflow boundary value problem for the Vlasov equation Asymptotic formulae involving quantum reflection transmission coefficients and time delays are used to construct a hybrid model coupling quantum descriptions to kinetic ones in a time dependent one dimensional setting 2000 Acad mie des sciences ditions scientifiques et m dicales Elsevier SAS Conditions aux limites entrantes pour l quation de Schr dinger unidimensionnelle dans le r gime transitoire R sum Un mod le transitoire de transport quantique monodimensionnel est propos L analyse asymptotique de ce mod le conduit dans la limite semi classique l quation de Vlasov avec condition au bord entrantes Des formules asymptotiques faisant intervenir les coefficients de r flexion transmission ainsi que les temps de retard quantiques sont obtenues et utilis es pour construire une mod le hybride cin tique quantique transitoire mondimensionnel 2000 Acad mie des sciences ditions scientifiques et m dicales Elsevier SAS Version fran aise abr g e



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