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

Abstract

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.

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

“…Since {γ (k) ∞,t } is obtained as a weak * limit of a BBGKY sequence, it verifes the assumption of the weak quantum de Finetti theorem in [32,9,29,18,40]. That…”

“…[37,39,33,34,35,1,2,4,5,6,7,8,20,22,23,24,21,10,18,26,28,36,31,30,25,29,32,38,40], and references therein. A fundamental problem is to prove that Bose-Einstein condensation occurs for such systems.…”

Higher Order Corrections to the Mean-Field Description of the Dynamics of Interacting Bosons

“…Subsequently, a crucial step of this method was revisited by Klainerman and Machedon in [33], based on reformulating combinatorial argument in [18,19] and a viewpoint inspired by methods of non-linear PDEs. This, in turn, motivated many recent works on the derivation of dispersive PDEs, including [11,12,13,14,15,32,53]. In [52], Rodnianski and Schlein introduced yet another method for proving (7), which uses coherent states on Fock space and was inspired by techniques of quantum field theory and the pioneering work of Hepp [29].…”

“…since the first double sum contributes only if 2n ≥ a + 1, and in this case min{2n, a} = a. Note that for k = n, j 1 = · · · = j k = 1, hence T With this, (53) and (54) imply for a = 0, 1 ψ(t) − ψ (1) ϕ (t) = T where we used that a + 2 ≤ 2a for a ≥ 2. To estimate t for j ∈ {1, 2} and any ψ ∈ L 2 sym (R dN ).…”

An infinite sequence of conserved quantities for the cubic Gross–Pitaevskii hierarchy on ℝ

“…Following [9], de Finetti theorems were successfully used to address unconditional uniqueness of certain GP hierarchies, see e.g. [50,36,37,17,33]. In particular, in the work at hand we use the unconditional uniqueness result for solutions to the cubic GP hierarchy in R d , d ≥ 1, obtained recently in [36].…”