Mathematical Sciences: The Wigner-Poisson and Related Systems

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9403458 Zweifel The Wigner Poisson System of equations describes electron plasmas in a regime where quantum effects are important, for example semi- conductor devices including field-effect transistors, Josephson junctions, hetero-junctions, quantum "dots" and the like. A self- consistent field approach is employed leading to pseudo-differential equations which in the classical limit tend to the ordinary Vlasov- Poisson system. The problem we propose to study includes existence and uniqueness of solutions, their regularity and asymptotic properties and developement of numerical methods appropriate to the treatment of practical devices, as well as studies of the classical limiting procedure. The studies to be carried out under this proposal are intended to lead to an understanding of methods applicable to many practical systems of interest in physics and engineering, especially systems of small size, because that is where quantum effects become important. Another important facet of our work will be to develop methods for obtaining quantum corrections to systems which are quasi-classical in nature, since such calculations will provide great economy in computer time and cost, vis-a-vis a full quantum description. Such calculations have been carried out by some researchers, but without any proof or even reasonable grounds to believe that such a procedure is mathematically valid.