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Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential

Authors: László Erdős, Benjamin Schlein and Horng-Tzer Yau
Journal: J. Amer. Math. Soc. 22 (2009), 1099-1156
MSC (2000): Primary 82C10, 35Q55
Published electronically: May 6, 2009
MathSciNet review: 2525781
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Abstract: Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\mathbf {x}=(x_1, \ldots , x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi _{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $\psi _{N,0}$ satisfies the energy condition \[ \langle \psi _{N,0}, H_N \psi _{N,0} \rangle \leq C N \] and that the one-particle density matrix converges to a projection as $N \to \infty$. Then, we prove that the $k$-particle density matrices of $\psi _{N,t}$ factorize in the limit $N \to \infty$. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant proportional to the scattering length of the potential $V$. In a recent paper, we proved the same statement under the condition that the interaction potential $V$ is sufficiently small. In the present work we develop a new approach that requires no restriction on the size of the potential.

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Additional Information

László Erdős
Affiliation: Institute of Mathematics, University of Munich, Theresienstrasse 39, D-80333 Munich, Germany
MR Author ID: 343945

Benjamin Schlein
Affiliation: DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom

Horng-Tzer Yau
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
MR Author ID: 237212

Received by editor(s): April 15, 2008
Published electronically: May 6, 2009
Additional Notes: The first author was partially supported by SFB/TR12 Project from DFG
The second author was supported by a Kovalevskaja Award from the Humboldt Foundation
The third author was partially supported by NSF grants DMS-0602038, 0757425, and 0804279
Article copyright: © Copyright 2009 American Mathematical Society