Abstract
We consider the Wigner equation corresponding to a nonlinear Schrödinger evolution of the Hartree type in the semiclassical limit h → 0. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L 2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L 2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which - as it is well known - is not pointwise positive in general.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 525-552 |
| Number of pages | 28 |
| Journal | Rendiconti Lincei - Matematica e Applicazioni |
| Volume | 22 |
| Issue number | 4 |
| DOIs | |
| State | Published - 2011 |
| Externally published | Yes |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: A. Athanassoulis would like to thank the CMLS, Ecole polytechnique for its hospitality during the preparation of this work. He was also partially supported by Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST). F. Pezzotti was partially supported by Project CBDif-Fr ANR-08-BLAN-0333-01.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.