Recent Results on the Cauchy Problem for Focusing and Defocusing Gross-Pitaevskii Hierarchies

Chen, Thomas; Pavlović, Nataša

doi:10.1051/mmnp/20105403

Cited by 7 publications

(9 citation statements)

References 25 publications

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“…A proof for the cases 0 < β ≤ 1 on a hierarchical method analogous to BBGKY hierarchies can be found in [1,[3][4][5]10]. However, these papers do not consider external field and do not estimate the error estimates on the convergence…”

Section: Introductionmentioning

confidence: 97%

Derivation of the time dependent Gross–Pitaevskii equation with external fields

Pickl¹

2015

Rev. Math. Phys.

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Using a new method [19], it is possible to derive mean field equations from the microscopic N body Schrödinger evolution of interacting particles without using BBGKY hierarchies. This method also allows for error estimates and can be generalized to systems with external fields, of which both points are relevant from a physics perspective.Recently, this method was used to derive the Hartree equation for singular interactions [11] and the Gross-Pitaevskii equation without positivity condition on the interaction [17] where one had to restrict the scaling behavior of the interaction.Assuming positivity of the interaction, this paper deals with more general scalings including the so-called Gross-Pitaevskii scaling.

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Section: Introductionmentioning

confidence: 97%

Derivation of the time dependent Gross–Pitaevskii equation with external fields

Pickl¹

2015

Rev. Math. Phys.

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“…The motivation is to study the Gross-Pitaevskii hierarchy as a generalization of the nonlinear Schrödinger equation via the factorized solutions and to prove analogues of the known results for the Cauchy problem for the NLS in the context of the GP hierarchy. By appropriately modifying the collision operator, it is also possible to consider a hierarchy which is related to the quintic NLS [22,23,24,25,26,28,29], as well as the NLS with more general power-type nonlinearities [107]. The Cauchy problem associated to the Hartree equation for infinitely many particles has recently been studied by Lewin and Sabin [82,83].…”

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confidence: 99%

A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on \( T^{3} \) from the dynamics of many-body quantum systems

Sohinger

2015

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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In this paper, we will obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on the three-dimensional torus T 3 from the many-body limit of interacting bosonic systems. This type of result was previously obtained on R 3 in the work of Erdős, Schlein, and Yau [51,52,53,54], and on T 2 and R 2 in the work of Kirkpatrick, Schlein, and Staffilani [75]. Our proof relies on an unconditional uniqueness result for the Gross-Pitaevskii hierarchy at the level of regularity α = 1, which is proved by using a modification of the techniques from the work of T. Chen, Hainzl, Pavlović and Seiringer [20] to the periodic setting. These techniques are based on the Quantum de Finetti theorem in the formulation of Ammari and Nier [6,7] and Lewin, Nam, and Rougerie [80]. In order to apply this approach in the periodic setting, we need to recall multilinear estimates obtained by Herr, Tataru, and Tzvetkov [71].Having proved the unconditional uniqueness result at the level of regularity α = 1, we will apply it in order to finish the derivation of the defocusing cubic nonlinear Schrödinger equation on T 3 , which was started in the work of Elgart, Erdős, Schlein, and Yau [47]. In the latter work, the authors obtain all the steps of Spohn's strategy for the derivation of the NLS [104], except for the final step of uniqueness. Additional arguments are necessary to show that the objects constructed in [47] satisfy the assumptions of the unconditional uniqueness theorem. Once we achieve this, we are able to prove the derivation result. In particular, we show Propagation of Chaos for the defocusing Gross-Pitaevskii hierarchy on T 3 for suitably chosen initial data. B j,k+1 (σ (k+1) ) := T r k+1 δ(x j − x k+1 ), σ (k+1) , 2010 Mathematics Subject Classification. 35Q55, 70E55.

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“…The case of singular convolution potentials was revisited in [41,42]. For a more detailed discussion on all of these results and further references, we refer the reader to [45, Subsection 1.3.2] and [71, Section 1.1 and Section 1.3], as well as to the expository works [19,69].…”

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confidence: 99%

The Gross–Pitaevskii Hierarchy on General Rectangular Tori

Herr

Sohinger

2015

Arch Rational Mech Anal

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Abstract. In this work, we study the Gross-Pitaevskii hierarchy on general -rational and irrational-rectangular tori of dimension two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in previous work of Kirkpatrick, Schlein, and Staffilani.

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Recent Results on the Cauchy Problem for Focusing and Defocusing Gross-Pitaevskii Hierarchies

Cited by 7 publications

References 25 publications

Derivation of the time dependent Gross–Pitaevskii equation with external fields

Derivation of the time dependent Gross–Pitaevskii equation with external fields

A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on \( T^{3} \) from the dynamics of many-body quantum systems

The Gross–Pitaevskii Hierarchy on General Rectangular Tori

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