Geodesic
A mathematical introduction to the Wigner formulation of quantum mechanics
Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 693-716.

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The paper is devoted to review, from a mathematical point of view, some fundamental aspects of the Wigner formulation of quantum mechanics. Starting from the axioms of quantum mechanics and of quantum statistics, we justify the introduction of the Wigner transform and eventually deduce the Wigner equation.
Il presente articolo è una rassegna di alcuni aspetti matematici fondamentali della formulazione Wigneriana della meccanica quantistica. A partire dagli assiomi della meccanica quantistica e della meccanica statistica quantistica viene motivata l'introduzione della trasformazione di Wigner e viene infine dedotta l'equazione di Wigner. 
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Barletti, Luigi. A mathematical introduction to the Wigner formulation of quantum mechanics. Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 3, pp. 693-716. http://geodesic.mathdoc.fr/item/BUMI_2003_8_6B_3_a13/

[1] A. Arnold-H. Lange-P. F. Zweifel, A discrete-velocity stationary Wigner equation, J. Math. Phys., 41 (2000), 7167-7180. | MR | Zbl

[2] A. Arnold-H. Steinrück, The "electromagnetic" Wigner equation for an electron with spin, Z. Angew. Math. Phys., 40 (1989), 793-815. | MR | Zbl

[3] J. Banasiak-L. Barletti, On the existence of propagators in stationary Wigner equation without velocity cut-off, Transport Theory Stat. Phys., 30 (2001), 659-672. | MR | Zbl

[4] L. Barletti-P. F. Zweifel, Parity-decomposition method for the stationary Wigner equation with inflow boundary conditions, Transport Theory Stat. Phys., 30 (2001), 507-520. | MR | Zbl

[5] N. Ben Abdallah-P. Degond-I. Gamba, Inflow boundary conditions for the time dependent one-dimensional Schrödinger equation, C. R. Acad. Sci. Paris, Sér. I Math., 331 (2000), 1023-1028. | MR | Zbl

[6] P. Billingsley, Probability and Measure (third edition), Wiley, 1995. | MR | Zbl

[7] P. Bordone-M. Pascoli-R. Brunetti-A. Bertoni-C. Jacoboni, Quantum transport of electrons in open nanostructures with the Wigner-function formalism, Phys. Rev. B, 59 (1999), 3060-3069.

[8] P. Carruthers-F. Zachariasen, Quantum collision theory with phase-space distributions, Rev. Mod. Phys., 55 (1983), 245-285. | MR

[9] T. A. Claasen-W. F. Mecklenbräuker, The Wigner distribution - a tool for time-frequency signal analysis, Philips J. Res., 35 (1980), 217-250. | MR | Zbl

[10] S. R. De Groot-L. G. Suttorp, Foundations of Electrodynamics, North-Holland, 1972.

[11] D. K. Ferry-S. M. Goodnick, Transport in Nanostructures, Cambridge University Press, 1997.

[12] R. P. Feynman, Statistical Mechanics, W. A. Benjamin Inc., 1972. | Zbl

[13] G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989. | MR | Zbl

[14] W. R. Frensley, Boundary conditions for open quantum systems driven far from equilibrium, Rev. Modern Phys., 62 (1990), 745-791.

[15] F. Frommlet, Time irreversibility in quantum mechanical systems, PhD thesis, Technischen Universität Berlin, 2000.

[16] P. Gérard-P. A. Markowich-N. J. Mauser-F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. | MR | Zbl

[17] R. L. Liboff, Kinetic Theory: Classical, Quantum and Relativistic Descriptions, Wiley, 1998.

[18] P. L. Lions-T. Paul, Sur les mesures de Wigner, Rev. Matematica Iberoamericana, 9 (1993), 553-618. | MR | Zbl

[19] G. W. Mackey, The Mathematical Foundations of Quantum Mechanics, W. A. Benjamin Inc., 1963. | Zbl

[20] P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations, Math. Meth. Appl. Sci., 11 (1989), 459-469. | MR | Zbl

[21] P. A. Markowich-N. J. Mauser-F. Poupaud, A Wigner function approach to (semi)classical limits: Electrons in a periodic potential, J. Math. Phys., 35 (1994), 1066-1094. | MR | Zbl

[22] P. A. Markowich-C. A. Ringhofer-C. Schmeiser, Semiconductor Equations, Springer Verlag, 1990. | MR | Zbl

[23] J. Von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. | MR | Zbl

[24] M. Reed-B. Simon, Methods of Modern Mathematical Physics, I - Functional Analysis, Academic Press, 1972. | MR | Zbl

[25] V. I. Tatarskiĭ, The Wigner representation of quantum mechanics, Sov. Phys. Usp., 26 (1983), 311-327. | MR

[26] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, 1950. | Jbk 58.1374.01 | Zbl

[27] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40 (1932), 749-759. | Jbk 58.0948.07

[28] P. Zhao-H. L. Cui-D. L. Woolard-K. L. Jensen-F. A. Buot, Equivalent circuit parameters of resonant tunneling diodes extracted from self-consistent Wigner-Poisson simulation, IEEE Transactions on Electron Devices, 48 (2001), 614-626.