Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons
| dc.creator | Elgart, Alexander | |
| dc.creator | Erdos, Laszlo | |
| dc.creator | Schlein, Benjamin | |
| dc.creator | Yau, Horng-Tzer | |
| dc.date | 2004-10-14 | |
| dc.date | 2004-10-26 | |
| dc.date.accessioned | 2026-07-07T04:31:32Z | |
| dc.date.available | 2026-07-07T04:31:32Z | |
| dc.description | We consider the dynamics of $N$ boson systems interacting through a pair potential $N^{-1} V_a(x_i-x_j)$ where $V_a (x) = a^{-3} V (x/a)$. We denote the solution to the $N$-particle Schrödinger equation by $ψ_{N, t}$. Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if $u_t$ solves the GP equation, then the family of $k$-particle density matrices $\{\otimes_k u_t, k\ge 1 \}$ solves the GP hierarchy. Under the assumption that $a = N^{-\eps}$ for $0 < \eps < 3/5$, we prove that as $N\to \infty$ the limit points of the $k$-particle density matrices of $ψ_{N,t}$ are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by $\int V(x) dx$. The uniqueness of the solutions to this hierarchy remains an open question. | |
| dc.description | Latex file, 18 pages | |
| dc.identifier | https://arxiv.org/abs/math-ph/0410038 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0410038 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/57851 | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35Q55; 81Q05; 81V70 | |
| dc.title | Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons | |
| dc.type | text |