Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate

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dc.creatorErdos, Laszlo
dc.creatorSchlein, Benjamin
dc.creatorYau, Horng-Tzer
dc.date2006-06-05
dc.date2006-12-10
dc.date.accessioned2026-07-07T07:16:55Z
dc.date.available2026-07-07T07:16:55Z
dc.descriptionConsider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $ψ_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $ψ_{N,0}$ satisfies the energy condition \[ < ψ_{N,0}, H_N^k ψ_{N,0} > \leq C^k N^k \] for $k=1,2,... $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to\infty$. We prove that the $k$-particle density matrices of $ψ_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic non-linear Schrödinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $ψ_{N,0}$ is assumed in a stronger sense.
dc.descriptionLatex file, 66 pages; new version, with an appendix to include a new class of inital states
dc.identifierhttps://arxiv.org/abs/math-ph/0606017
dc.identifierhttp://arxiv.org/abs/math-ph/0606017
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/113760
dc.subjectMathematical Physics
dc.subject35Q55; 81Q15; 81T18; 81V70
dc.titleDerivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate
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