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The Wigner formulation of quantum mechanics [1] has some features that makes it a good alternative to the Schrödinger formulation to several respects:
1. it allows to investigate the semi-classical limit of a quantum system in a particularly clear fashion;
2. it presents striking formal analogies with the classical theory of kinetic equations;
3. it suggest the form of quantum versions of Boltzmann or Vlasov equations. However, these properties have to be taken very carefully, since the analogy with the kinetic theory is seriously restricted by the fact that the Wigner transform of a wave function is not a density in phase-space as it may take negative values. This fact has some important consequences in the study of Wigner equation in a bounded domain, making very unclear the formulation of physically reasonable boundary conditions. On the other hand, the study of Wigner or Wigner-Poisson equations in a bounded domain has a remarkable interest for the applications to mathematical modeling of semiconductor devices, where the bunded domain has to represent the functional site of the device, while boundary conditions have to account for the presence of ohmic contacts [2].
So far, both the stationary and non-stationary Wigner equation in a bounded domain has been investigated with a semi-classical ``inflow'' boundary condition, under some technical restriction on the choice of the momentum of particles [3]. Our research project in this field can be divided in three branches:
1. theoretical study of the stationary Wigner (linear) and Wigner-Poisson (nonlinear) equation in a bounded domain with inflow boundary conditions;
2. theoretical study of alternative boundary conditions for the Wigner equation in bounded domain;
3. application to the mathematical modeling of an IRTD device (see below).
Next: Wigner
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