Far from equilibrium steady states of 1D-Schrödinger–Poisson systems with quantum wells I

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].