Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy

Abstract

We consider the dynamics of $N$ bosons in three dimensions. We assume the pair interaction is given by $N^{3\beta -1}V(N^{\beta }\cdot )$ . By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the $N$-body wave function, for $\beta \in \left(0,1\right]$, we derive the Gross-Pitaevskii hierarchy with $2$-body interaction. In particular, we establish that, in the $N\to \infty$ limit, all $k$-body scattering processes vanishes if $k⩾3$ and thus provide a direct answer to a question raised by Erdös, Schlein, and Yau in [31]. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for $\beta \in \left(0,1\right)$ and weakly for $\beta =1$. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three body estimate for which we have the estimates to achieve $\beta <1$, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman and Machedon in [47] for $\beta \in \left(0,1\right)$.

References in corpus (5)

• Superfluid behaviour of a two-dimensional Bose gas
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy
• Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D

Cited by in corpus (15)

• Gross-Pitaevskii Dynamics for Bose-Einstein Condensates
• Complete Bose-Einstein condensation in the Gross-Pitaevskii regime
• Pair excitations and the mean field approximation of interacting Bosons, II
• Global Well-posedness of the NLS System for infinitely many fermions
• The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
• The Derivation of the ${\mathbb{T}}^3$ Energy-critical NLS from Quantum Many-body Dynamics
• Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D
• Central limit theorem for Bose gases interacting through singular potentials
• The rigorous derivation of the ${\mathbb{T}}^2$ focusing cubic NLS from 3D
• Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space
• The unconditional uniqueness for the energy-supercritical NLS
• The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$
• Well/Ill-posedness of the Boltzmann Equation with Soft Potential
• Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel
• On the mean-field and semiclassical limit from quantum $N$-body dynamics