Unconditional Uniqueness for the Cubic Gross‐Pitaevskii Hierarchy via Quantum de Finetti

Chen, Thomas; Hainzl, Christian; Pavlović, Nataša; Seiringer, Robert

doi:10.1002/cpa.21552

Cited by 65 publications

(199 citation statements)

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“…The Cauchy problem associated to a hierarchy as in (1) has been studied in its own right in [41][42][43][44]46,47,59]. A new unconditional uniqueness result for the cubic Gross-Pitaevskii hierarchy on R 3 , based on the Quantum de Finetti theorem has recently been obtained by Chen, Hainzl, Pavlović, and Seirenger in [39]. These techniques have been adapted in order to show scattering results in the context of the Gross-Pitaevskii hierarchy in [40].…”

Section: Previously Known Resultsmentioning

confidence: 99%

“…Low regularity extensions of this method have been obtained in [103], as well as in the case of three-body interactions [104]. The periodic analogue of the methods from [39] was an important step in [147]. Recently, the Quantum de Finetti was also used in the study of the Chern-Simons-Schrödinger hierarchy in [58].…”

Section: Previously Known Resultsmentioning

confidence: 99%

“…In our previous joint work with Gressman,[92], we proved a conditional uniqueness result for the Gross-Pitaevskii hierarchy on T 3 , which is the 3D analogue of the uniqueness result used in [107] for regularity strictly greater than 1. In a recent paper, Chen, Hainzl, Pavlović, and Seiringer [39] used the Quantum de Finetti theorem and gave an alternative proof of the uniqueness result on R 3 from [77]. An extension of the above results to the context of general rectangular tori was given in the first author's joint work with Herr [102].…”

Section: Setup Of the Problemmentioning

confidence: 99%

“…In this subsection, we construct factorized solutions to (39). As we will see, they will be obtained as tensor products of solutions to a nonlinear Schrödinger equation with a random nonlinearity.…”

Section: Link With the Nonlinear Schrödinger Equationmentioning

confidence: 99%

“…We can now use the function ψ ω to find a factorized solution to the randomized Gross-Pitaevskii hierarchy (39). In particular, if we take initial data γ (39), as long as one works in a class of permutation symmetric density matrices.…”

Section: Link With the Nonlinear Schrödinger Equationmentioning

confidence: 99%

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Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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We study the Gross-Pitaevskii hierarchy on the spatial domain T 3 . By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is α > 3 4 . It was shown in our previous joint work with Gressman (J Funct Anal 266(7): 2014) that the range α > 1 is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon (Commun Math Phys 279(1): [169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184][185] 2008), where the spacetime estimate is known to hold whenever α ≥ 1. The goal of our paper is to extend the range of α in this class of estimates in a probabilistic sense. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

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Section: Previously Known Resultsmentioning

confidence: 99%

Section: Previously Known Resultsmentioning

confidence: 99%

Section: Setup Of the Problemmentioning

confidence: 99%

Section: Link With the Nonlinear Schrödinger Equationmentioning

confidence: 99%

Section: Link With the Nonlinear Schrödinger Equationmentioning

confidence: 99%

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Randomization and the Gross–Pitaevskii Hierarchy

Sohinger

Staffilani

2015

Arch Rational Mech Anal

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Effective Evolution Equations from Quantum Dynamics

Benedikter¹,

Porta²,

Schlein³

2016

SpringerBriefs in Mathematical Physics

131

221

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In these notes we review the material presented at the summer school on "Mathematical Physics, Analysis and Stochastics" held at the University of Heidelberg in July 2014. We consider the time-evolution of quantum systems and in particular the rigorous derivation of effective equations approximating the many-body Schrödinger dynamics in certain physically interesting regimes.for an appropriate constant C > 0. This gives a mathematically rigorous derivation of the Thomas-Fermi theory, and it tells us how big N should be in order for E T F (N ) to be a good approximation of the true quantum energy (later, better bounds have been obtained [41,58,59]: one knows that the error with respect to Thomas-Fermi theory is of the order N 2 in the limit of large N ).2 Mean-field regime for bosonic systems

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