On Wigner measures.

Pierre-Louis Lions; Thierry Paul

Revista Matemática Iberoamericana (1993)

  • Volume: 9, Issue: 3, page 553-618
  • ISSN: 0213-2230

Abstract

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We study the properties of the Wigner transform for arbitrary functions in L2 or for hermitian kernels like the so-called density matrices. And we introduce some limits of these transforms for sequences of functions in L2, limits that correspond to the semi-classical limit in Quantum Mechanics. The measures we obtain in this way, that we call Wigner measures, have various mathematical properties that we establish. In particular, we prove they satisfy, in linear situations (Schrödinger equations) or nonlinear ones (time-dependent Hartree equations), transport equations of Liouville or Vlasov type.

How to cite

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Lions, Pierre-Louis, and Paul, Thierry. "Sur les mesures de Wigner.." Revista Matemática Iberoamericana 9.3 (1993): 553-618. <http://eudml.org/doc/39445>.

@article{Lions1993,
author = {Lions, Pierre-Louis, Paul, Thierry},
journal = {Revista Matemática Iberoamericana},
keywords = {Mecánica cuántica; Mecánica estadística; Transformada de Wigner; Ecuación de Liouville; Métodos fisicomatemáticos; Medidas; Schrödinger equation; Wigner transform; semi-classical limit; Hartree equation},
language = {fre},
number = {3},
pages = {553-618},
title = {Sur les mesures de Wigner.},
url = {http://eudml.org/doc/39445},
volume = {9},
year = {1993},
}

TY - JOUR
AU - Lions, Pierre-Louis
AU - Paul, Thierry
TI - Sur les mesures de Wigner.
JO - Revista Matemática Iberoamericana
PY - 1993
VL - 9
IS - 3
SP - 553
EP - 618
LA - fre
KW - Mecánica cuántica; Mecánica estadística; Transformada de Wigner; Ecuación de Liouville; Métodos fisicomatemáticos; Medidas; Schrödinger equation; Wigner transform; semi-classical limit; Hartree equation
UR - http://eudml.org/doc/39445
ER -

Citations in EuDML Documents

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  1. Guillaume Bal, Lenya Ryzhik, Time splitting for wave equations in random media
  2. Marjolaine Puel, Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field
  3. I. Gasser, R. Illner, P. A. Markowich, C. Schmeiser, Semiclassical, t → ∞ asymptotics and dispersive effects for Hartree-Fock systems
  4. Luigi Barletti, A mathematical introduction to the Wigner formulation of quantum mechanics
  5. Marcello Porta, Mean-field evolution of fermionic systems
  6. Marjolaine Puel, Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field
  7. Guillaume Bal, Lenya Ryzhik, Time splitting for wave equations in random media
  8. Christophe Gomez, Olivier Pinaud, Asymptotics of a Time-Splitting Scheme for the Random Schrödinger Equation with Long-Range Correlations
  9. Olof Runborg, Some new results in multiphase geometrical optics
  10. Patrick Gérard, Résultats de propagation pour les équations aux dérivées partielles à coefficients oscillants

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