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The stationary one dimensional Schr\"odinger-Poisson system on a boundedinterval is considered in the limit of a small Debye length (or smalltemperature). Electrons are supposed to be in a mixed state with the Boltzmannstatistics. Using various reformulations of the system as convex minimizationproblems, we show that only the first energy level is asymptotically occupied.The electrostatic potential is shown to converge towards a boundary layerpotential with a profile computed by means of a half spaceSchr\"odinger-Poisson system.

High Density Limit of the Stationary One Dimensional Schrödinger–Poisson System