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The Wigner Poisson problem in a crystal: Existence, uniqueness, semiclassical limit in the one-dimensional case. (English) Zbl 0763.35094

Summary: We analyse a quantum-mechanical model for the transport of electrons in a semiconductor. The model consists of the Wigner equation posed on the bounded Brillouin zone corresponding to the crystal lattice of the semiconductor, with a selfconsistent potential determined by a discrete Poisson equation. We prove global existence and uniqueness of solutions and consider the semiclassical limit.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35Q05 Euler-Poisson-Darboux equations
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
82C70 Transport processes in time-dependent statistical mechanics
82D25 Statistical mechanics of crystals
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