Propagation of chaos for many-boson systems in one dimension with a point pair-interaction

Abstract

We consider the semiclassical limit of nonrelativistic quantum many-boson systems with delta potential in one dimensional space. We prove that time evolved coherent states behave semiclassically as squeezed states by a Bogoliubov time-dependent affine transformation. This allows us to obtain properties analogous to those proved by Hepp and Ginibre-Velo ([Hep], [GiVe1, GiVe2]) and also to show propagation of chaos for Schrödinger dynamics in the mean field limit. Thus, we provide a derivation of the cubic NLS equation in one dimension.

On the mean-field approximation of many-boson dynamics

“…Moreover, dΓ(A) + N has an invariant form domain with respect to the Weyl operator W(ξ) when ξ ∈ Q(A). This propriety can be proved using the Faris-Lavine argument [23] and it is proved for instance in [3]. Then the Wigner measure µ is carried by Q(A) (i.e.…”

Gross–Pitaevskii dynamics for Bose–Einstein condensates

“…where we isolated the terms with (n, k) = (0, 0) and (n, k) = (0, 1) and the sum * runs over all other pairs (n, k) ∈ N × N. The first term on the r.h.s. of (5.92) (the one associated with (k, n) = (0, 0)) is subtracted in (5.91) and does not enter the error term E (2) N,t . The second term on the r.h.s.…”

Bohr's Correspondence Principle for the Renormalized Nelson Model

“…The "dressed" coupling: The system that arises from the dressed interaction is quite complicated. We will denote it by S-KG[D], and it has the following form 6 :…”

“…) the unknowns are u and α. 6 We denote by ∂ (i) the derivative with respect to the i-th component of the variable x ∈ R 3 . Analogously, we denote by v (i) the i-th component of a 3-dimensional vector v.…”