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.).


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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