Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics

Abstract

We derive rigorously, for both R 2 and [−L, L] ×2 , the cubic nonlinear Schrödinger equation in a suitable scaling limit from the two-dimensional many-body Bose systems with short-scale repulsive pair interactions. We first prove convergence of the solution of the BBGKY hierarchy, corresponding to the many-body systems, to a solution of the infinite Gross-Pitaevskii hierarchy, corresponding to the cubic NLS; and then we prove uniqueness for the infinite hierarchy, which requires number-theoretical techniques in the periodic case.

Beyond mean field: On the role of pair excitations in the evolution of condensates

“…The strategy of these papers is based on the older work of Spohn [30]. Recent simplifications and generalizations, based on harmonic analysis techniques and a "boardgame argument" inspired by the Feynman diagram approach of Erdös, Schlein and Yau, were given in [21], [22], [6], [3], [4], [5]. See also [14], [27] for a different approach.…”

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

“…holds, with C independent of k. The authors of [22] proved that the latter is indeed satisfied for the cubic case in d = 2, where energy conservation in the N -particle system is shown to imply that B j;k+1 γ (k+1) Ḣ1 k < C k , even without invoking the norm in the time variable. In [6] we proved the analogous bound for the quintic case in d = 1, 2.…”

“…For the approach developed in [21], the authors make the assumption of a particular a priori spacetime bound on the density matrices. In the work [22] of Kirkpatrick, Schlein, and Staffilani, the corresponding problem in d = 2 is solved, and the assumption made in [21] is replaced by a spatial a priori bound which is proven in [22].…”

“…In order to prove the uniqueness of the limiting hierarchy, we expand the approach introduced by Klainerman-Machedon [21] and subsequently used by Kirkpatrick-Schlein-Staffilani [22]. Thereby, we obtain a mathematically rigorous derivation of the quintic NLS.…”

“…For some fundamental results in this field, we refer to the works of Erdös, Schlein and Yau in [12,13,14] which have initiated much of the current interest in this area. See also [21,22,26] and the references therein; see also [1,3,11,15,16,17,18,19,20,28].…”