On Wigner measures. (Sur les mesures de Wigner.) (French) Zbl 0801.35117

Lions, Pierre Louis; Paul, Thierry

Rev. Mat. Iberoam. 9, No. 3, 553-618 (1993).

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.).

Reviewer: M.Perelmuter (Kiev)

MSC:

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

Keywords:

Schrödinger equation; Wigner transform; semi-classical limit; Hartree equation