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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential
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by László Erdős, Benjamin Schlein and Horng-Tzer Yau
J. Amer. Math. Soc. 22 (2009), 1099-1156
Published electronically: May 6, 2009


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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Bibliographic 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
  • © Copyright 2009 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 22 (2009), 1099-1156
  • MSC (2000): Primary 82C10, 35Q55
  • DOI:
  • MathSciNet review: 2525781