Far from equilibrium steady states of 1D-Schrödinger–Poisson systems with quantum wells I
Y. Patel
IRMAR, UMR–CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes, FranceV. Bonnaillie-Noël
IRMAR, UMR–CNRS 6625, ENS Cachan Bretagne, Université Rennes 1, UEB, av. Robert Schuman, 35170 Bruz, FranceF. Nier
IRMAR, UMR–CNRS 6625, Université Rennes 1, Campus de Beaulieu, 35042 Rennes, France
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Abstract
We describe the asymptotics of the steady states of the out-of-equilibrium Schrödinger–Poisson system, in the regime of quantum wells in a semiclassical island. After establishing uniform estimates on the nonlinearity, we show that the nonlinear steady states lie asymptotically in a finite-dimensional subspace of functions and that the involved spectral quantities are reduced to a finite number of so-called asymptotic resonant energies. The asymptotic finite dimensional nonlinear system is written in a general setting with only a partial information on its coefficients. After this first part, a complete derivation of the asymptotic nonlinear system will be done for some specific cases in a forthcoming article [V. Bonnaillie–Noël, F. Nier, M. Patel, Far from equilibrium steady states of 1D-Schrödinger–Poisson systems with quantum wells II, Prépublications IRMAR, 2007].
Cite this article
Y. Patel, V. Bonnaillie-Noël, F. Nier, Far from equilibrium steady states of 1D-Schrödinger–Poisson systems with quantum wells I. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 5, pp. 937–968
DOI 10.1016/J.ANIHPC.2007.05.007