## Asymptotic analysis of a scaled Wigner equation and quantum scattering.(English)Zbl 0870.45003

Summary: We show how a scaling on the classical or quantum Liouville equation leads to scattering theory. After a preliminary analysis of classical mechanics, we focus on the quantum case treated via the Wigner function approach. The problem arises from a one-dimensional modelling of electron transport in quantum electronic devices and is more generally related to the study of scattering into cones.

### MSC:

 45K05 Integro-partial differential equations 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 45M05 Asymptotics of solutions to integral equations 81U20 $$S$$-matrix theory, etc. in quantum theory
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