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

Authors: László Erdos, 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

$\displaystyle \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 Erdos
Affiliation: Institute of Mathematics, University of Munich, Theresienstrasse 39, D-80333 Munich, Germany

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

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