# On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Journal of the European Mathematical Society (2016)

- Volume: 018, Issue: 6, page 1161-1200
- ISSN: 1435-9855

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topChen, Xuwen, and Holmer, Justin. "On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction." Journal of the European Mathematical Society 018.6 (2016): 1161-1200. <http://eudml.org/doc/277721>.

@article{Chen2016,

abstract = {We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^\{3\beta – 1\} V (B^\beta )$. For the interaction parameter $\beta \in (0, 2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N \rightarrow \infty $ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0, 3/5)$ in [28]. This allows, in the case $\beta \in (0, 3/5)$, for the application of the Klainerman–Machedon uniqueness theorem and hence implies that the $N \rightarrow \infty $ limit of BBGKY is uniquely determined as a tensor product of solutions to the Gross–Pitaevskii equation when the $N$-body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlović [11] for $\beta \in (0, 1/4)$ and subsequently by X. Chen [15] for $\beta \in (0, 2/7)$. We build upon the approach of X. Chen but apply frequency localized Klainerman–Machedon collapsing estimates and the endpoint Strichartz estimate in the estimate of the “potential part” to extend the range to $\beta \in (0, 2/3)$. Overall, this provides an alternative approach to the mean-field program by L. Erdős, B. Schlein, and H.-T. Yau [28], whose uniqueness proof is based upon Feynman diagram combinatorics.},

author = {Chen, Xuwen, Holmer, Justin},

journal = {Journal of the European Mathematical Society},

keywords = {BBGKY hierarchy; $n$-particle Schrödinger Equation; Klainerman–Machedon space-time bound; quantum Kac program; BBGKY hierarchy; $n$-particle Schrödinger equation; klainerman-machedon space-time bound; quantum Kac program},

language = {eng},

number = {6},

pages = {1161-1200},

publisher = {European Mathematical Society Publishing House},

title = {On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction},

url = {http://eudml.org/doc/277721},

volume = {018},

year = {2016},

}

TY - JOUR

AU - Chen, Xuwen

AU - Holmer, Justin

TI - On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

JO - Journal of the European Mathematical Society

PY - 2016

PB - European Mathematical Society Publishing House

VL - 018

IS - 6

SP - 1161

EP - 1200

AB - We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta – 1} V (B^\beta )$. For the interaction parameter $\beta \in (0, 2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N \rightarrow \infty $ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0, 3/5)$ in [28]. This allows, in the case $\beta \in (0, 3/5)$, for the application of the Klainerman–Machedon uniqueness theorem and hence implies that the $N \rightarrow \infty $ limit of BBGKY is uniquely determined as a tensor product of solutions to the Gross–Pitaevskii equation when the $N$-body initial data is factorized. The first result in this direction in 3D was obtained by T. Chen and N. Pavlović [11] for $\beta \in (0, 1/4)$ and subsequently by X. Chen [15] for $\beta \in (0, 2/7)$. We build upon the approach of X. Chen but apply frequency localized Klainerman–Machedon collapsing estimates and the endpoint Strichartz estimate in the estimate of the “potential part” to extend the range to $\beta \in (0, 2/3)$. Overall, this provides an alternative approach to the mean-field program by L. Erdős, B. Schlein, and H.-T. Yau [28], whose uniqueness proof is based upon Feynman diagram combinatorics.

LA - eng

KW - BBGKY hierarchy; $n$-particle Schrödinger Equation; Klainerman–Machedon space-time bound; quantum Kac program; BBGKY hierarchy; $n$-particle Schrödinger equation; klainerman-machedon space-time bound; quantum Kac program

UR - http://eudml.org/doc/277721

ER -

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