7,102 research outputs found
On the Klainerman-Machedon Conjecture of the Quantum BBGKY Hierarchy with Self-interaction
We consider the 3D quantum BBGKY hierarchy which corresponds to the
-particle Schr\"{o}dinger equation. We assume the pair interaction is
For interaction parameter , we prove that, as the limit points of
the solutions to the BBGKY hierarchy satisfy the space-time bound conjectured
by Klainerman-Machedon in 2008. This allows for the application of the
Klainerman-Machedon uniqueness theorem, and hence implies that the limit is
uniquely determined as a tensor product of solutions to the Gross-Pitaevski
equation when the -body initial data is factorized. The first result in this
direction in 3D was obtained by T. Chen and N. Pavlovi\'{c} (2011) for and subsequently by X. Chen (2012) for .
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 . Overall, this provides an alternative approach to the mean-field
program by Erd\"os-Schlein-Yau (2007), whose uniqueness proof is based upon
Feynman diagram combinatorics.Comment: v2, final version for Journal of the European Mathematical Society.
v1 is a less technical versio
The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
We consider a 2D time-dependent quantum system of -bosons with harmonic
external confining and \emph{attractive} interparticle interaction in the
Gross-Pitaevskii scaling. We derive stability of matter type estimates showing
that the -th power of the energy controls the Sobolev norm of the
solution over -particles. This estimate is new and more difficult for
attractive interactions than repulsive interactions. For the proof, we use a
version of the finite-dimensional quantum di Finetti theorem from [49]. A high
particle-number averaging effect is at play in the proof, which is not needed
for the corresponding estimate in the repulsive case. This a priori bound
allows us to prove that the corresponding BBGKY hierarchy converges to the GP
limit as was done in many previous works treating the case of repulsive
interactions. As a result, we obtain that the \emph{focusing} nonlinear
Schr\"{o}dinger equation is the mean-field limit of the 2D time-dependent
quantum many-body system with attractive interatomic interaction and
asymptotically factorized initial data. An assumption on the size of the
-norm of the interatomic interaction potential is needed that
corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality
though the inequality is not directly relevant because we are dealing with a
trace instead of a power
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