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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