# Mean field limit for bosons with compact kernels interactions by Wigner measures transportation

## Abstract

We consider a class of many-body Hamiltonians composed of a free (kinetic) part and a multi-particle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove in particular the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in Z. Ammari and F. Nier and Fröhlich et al. [“Mean field limit for bosons and propagation of Wigner measures,” J. Math. Phys. 50(4), 042107 (2009); Z. Ammari and F. Nier and Fröhlich et al., “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states,” J. Math. Pures Appl. 95(6), 585–626 (2011); Z. Ammari and F. Nier and Fröhlich et al., “Mean-field- and classical limit of many-body Schrödinger dynamics for bosons,” Commun. Math. Phys. 271(3), 681–697 (2007)].

- Authors:

- IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex (France)

- Publication Date:

- OSTI Identifier:
- 22306037

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 9; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BBGKY EQUATION; BOSONS; FIELD EQUATIONS; HAMILTONIANS; MANY-BODY PROBLEM; MEAN-FIELD THEORY; NONLINEAR PROBLEMS

### Citation Formats

```
Liard, Quentin, E-mail: quentin.liard@univ-rennes1.fr, and Pawilowski, Boris, E-mail: boris.pawilowski@univ-rennes1.fr.
```*Mean field limit for bosons with compact kernels interactions by Wigner measures transportation*. United States: N. p., 2014.
Web. doi:10.1063/1.4895467.

```
Liard, Quentin, E-mail: quentin.liard@univ-rennes1.fr, & Pawilowski, Boris, E-mail: boris.pawilowski@univ-rennes1.fr.
```*Mean field limit for bosons with compact kernels interactions by Wigner measures transportation*. United States. doi:10.1063/1.4895467.

```
Liard, Quentin, E-mail: quentin.liard@univ-rennes1.fr, and Pawilowski, Boris, E-mail: boris.pawilowski@univ-rennes1.fr. Mon .
"Mean field limit for bosons with compact kernels interactions by Wigner measures transportation". United States. doi:10.1063/1.4895467.
```

```
@article{osti_22306037,
```

title = {Mean field limit for bosons with compact kernels interactions by Wigner measures transportation},

author = {Liard, Quentin, E-mail: quentin.liard@univ-rennes1.fr and Pawilowski, Boris, E-mail: boris.pawilowski@univ-rennes1.fr},

abstractNote = {We consider a class of many-body Hamiltonians composed of a free (kinetic) part and a multi-particle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove in particular the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in Z. Ammari and F. Nier and Fröhlich et al. [“Mean field limit for bosons and propagation of Wigner measures,” J. Math. Phys. 50(4), 042107 (2009); Z. Ammari and F. Nier and Fröhlich et al., “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states,” J. Math. Pures Appl. 95(6), 585–626 (2011); Z. Ammari and F. Nier and Fröhlich et al., “Mean-field- and classical limit of many-body Schrödinger dynamics for bosons,” Commun. Math. Phys. 271(3), 681–697 (2007)].},

doi = {10.1063/1.4895467},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 9,

volume = 55,

place = {United States},

year = {2014},

month = {9}

}