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On Wigner measures. (Sur les mesures de Wigner.) (French) Zbl 0801.35117
The authors investigate the properties of the Wigner transform for arbitrary functions in \(L^ 2\): \[ W(x,\xi)= (2\pi)^{-2N} \int_{\mathbb{R}^ N} \exp(-i\xi y)\psi(x+ y/2)\psi^*(x- y/2)dy,\;(x,\xi)\in \mathbb{R}^ N\times \mathbb{R}^ N \] and Wigner transform for density matrices: \[ \rho(x,y)= \psi(x)\psi^*(y),\;W(x,\xi)= (2\pi)^{-2N} \int_{\mathbb{R}^ N} \exp(-i\xi y)\rho(x+ y/2,x- y/2)dy. \] Some limits of these transforms for sequences of functions are introduced. These limits correspond to the semi-classical limit in quantum mechanics. The various properties of these limits are obtained. In particular, they satisfy some classical equations (Schrödinger equation, Hartree equation, etc.).

MSC:
35Q40 PDEs in connection with quantum mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
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