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On a one-dimensional Schrödinger-Poisson scattering model. (English) Zbl 0885.34067
The authors consider the so-called Schrödinger-Poisson model for describing the stationary states of a quantum device. This is a model of the charged particle transport in semiconductor devices. In dimensionless units, the electronic state in the device is modeled on the domain \((0,1)\) by the wave function \(\Psi_k\) satisfying the Schrödinger equation, \(-h^2\Psi''_k+ V\Psi_k= k^2\Psi_k\), with the boundary conditions \(h\Psi_k'(0)+ ik\Psi_k(0)= 2ik\) and \(h\Psi_k'(1)= i\sqrt{k^2- V_1}\Psi_k(1)\). Here, the electric potential \(V\) is a solution of the Poisson equation on \((0,1)\), \(-V''(x)= n(x)\), with the boundary conditions \(V(0)= 0\) and \(V(1)= V_1\), where \(V_1\) is an external voltage.
The authors prove the existence of solutions to this model. In the semiclassical limit performed via a Wigner transformation the model is reduced to the standard boundary value problem for the semiclassical Vlasov-Poisson system.

MSC:
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81U10 \(n\)-body potential quantum scattering theory
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
78A35 Motion of charged particles
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