Derivation of the Gross-Pitaevskii Hierarchy for the Dynamics of Bose-Einstein Condensate
| dc.creator | Erdos, Laszlo | |
| dc.creator | Schlein, Benjamin | |
| dc.creator | Yau, Horng-Tzer | |
| dc.date | 2004-10-02 | |
| dc.date | 2005-10-19 | |
| dc.date.accessioned | 2026-07-07T06:38:32Z | |
| dc.date.available | 2026-07-07T06:38:32Z | |
| dc.description | Consider a system of $N$ bosons on the three dimensional unit torus interacting via a pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, ..., x_N)$ denotes the positions of the particles. Suppose that the initial data $ψ_{N,0}$ satisfies the condition \[ < ψ_{N,0}, H_N^2 ψ_{N,0} > \leq C N^2 \] where $H_N$ is the Hamiltonian of the Bose system. This condition is satisfied if $ψ_{N,0}= W_N ϕ_{N,0}$ where $W_N$ is an approximate ground state to $H_N$ and $ϕ_{N,0}$ is regular. Let $ψ_{N,t}$ denote the solution to the Schrödinger equation with Hamiltonian $H_N$. Gross and Pitaevskii proposed to model the dynamics of such system by a nonlinear Schrödinger equation, the Gross-Pitaevskii (GP) equation. 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. 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. The uniqueness of the solutions to this hierarchy remains an open question. Our analysis requires that the $N$ boson dynamics is described by a modified Hamiltonian which cuts off the pair interactions whenever at least three particles come into a region with diameter much smaller than the typical inter-particle distance. Our proof can be extended to a modified Hamiltonian which only forbids at least $n$ particles from coming close together, for any fixed $n$. | |
| dc.description | Latex file, 66 pages | |
| dc.identifier | https://arxiv.org/abs/math-ph/0410005 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0410005 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100769 | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35Q55; 81Q05; 81V70 | |
| dc.title | Derivation of the Gross-Pitaevskii Hierarchy for the Dynamics of Bose-Einstein Condensate | |
| dc.type | text |