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.


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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[1] Agmon S., Ann. Sc. Norm. Sup. Pisa 4 pp 151– (1975)
[2] Agmon S., Some New Results in Spectral and Scattering Theory of Differential Operators on Rn (1978)
[3] Amrein W. O., Scattering Theory in Quantum Mechanics (1977) · Zbl 0376.47001
[4] DOI: 10.1090/S0025-5718-1992-1122055-5
[5] Balian R., Statistique (1986)
[6] DOI: 10.1007/978-1-4612-1039-9
[7] Cohen-Tannoudji C., Mécanique Quantique 2 (1973)
[8] Deift P., CPAM pp 32– (1979)
[9] gond De P., J. Comp. Phys. 78 (1988)
[10] DOI: 10.1007/BF01661573
[11] Frensley W. R., Phys. Rev. B pp 36– (1987)
[12] Harris S., An introduction to the Theory of the Boltzmann Equation (1977)
[13] DOI: 10.1002/cpa.3160320304 · Zbl 0388.47032
[14] DOI: 10.1007/BF01646269 · Zbl 0159.55001
[15] Isozaki H., Sci. Papers College Arts Sci. Univ. Tokyo 35 pp 81– (1985)
[16] Lions, P. L. and Paul, T. 1992.Sur les mesures de Wigner, Vol. IX, 9227Cahiers du CEREMADE Université Paris. · Zbl 0801.35117
[17] DOI: 10.1002/mma.1670110404 · Zbl 0696.47042
[18] Messiah A., Mécanique Quantique (1965)
[19] Newton R. G., Scattering Theory of Waves and Particles (1966)
[20] Nier, F. 1991. ”Etude mathématique et numérique de modèles cinétiques quantiques issus de la physique des semi-conducteurs”. Palaiseau, France: Thèse de L’Ecole Polytechnique.
[21] DOI: 10.1063/1.1665952 · Zbl 0239.70004
[22] Ravaioli U., Physica 134 pp 36– (1985)
[23] Reed M., Methods of Modern Mathematical Physics (1975) · Zbl 0308.47002
[24] DOI: 10.1080/00411458908204692 · Zbl 0703.35163
[25] Robert D., Autour del’approximation semi-classique (1987)
[26] DOI: 10.1070/PU1983v026n04ABEH004345
[27] Thirring W., A Course in Mathematical Physics 3: Quantum Mechanics of Atoms and Molecules (1981) · Zbl 0462.46046
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